A period/full stop can be used to indicate a syllable break, but it isn't required or necessarily preferred to do so. (Similarly, words often are not written with spaces between them.)

A colon, however, is an easy/lazy typographical variant of the long vowel symbol "ː".

I have read that forward slashes are used for phonetic translations and narrow transcriptions, whereas brackets are used for phonemic translation and broad transcriptions. Is this correct?

Yes, that is correct. The two different formats have different purposes.

If you are describing the lexical form of a word, as phonemes, use slashes. This is called a broad transcription.

If you are describing a specific phonetic pronunciation, after phonological changes have occurred, use brackets. This is called a narrow transcription. Note that there are different degrees of narrow transcriptions, but if you are using anything other than a broad phonemic transcription, you probably should consider it narrow.

As a very basic example, consider the phoneme /s/ in English, which has frequent allophones [s] and [z].

Yes. /s/ is a phoneme, and [s] is one possible pronunciation of that phoneme.

If you're citing the lexical/underlying/phonemic/broad form of the word, then slashes should be used. However, it is also possible that is actually intended as the narrow pronunciation. If it is possible to pronounce the underlying form without any relevant phonological rules taking place, at the level of detail you're using for narrow transcription, then both could be correct.

Usually there are some differences. For example, the phoneme /k/ is usually aspirated word-initially in English as [kʰ]. Other changes might also apply depending on the phonological rules you've established and contextual effects. But perhaps this would be the way to write it:

/kʌrɪdʒ/
[kʰʌrɪdʒ]

Narrowness really does have multiple levels, down to the details of things like fast speech or even speech errors. So in the right situation you might also have as a pronunciation, monosyllabification:
[kʰʌrdʒ]
That's not a standard phonological rule, just something that might happen when speaking quickly. That would also go in square brackets as a specific pronunciation.

The idea behind // vs. [] is very simple, but in practice it depends on the level of detail you're using, and conventions.

Another example:
/kʌrɪdʒ/
vs.
[kʰʌɹɪdʒ]

The first uses the phoneme /r/ (there's no need to write any other contrastive symbol there, just "R", because English only has one), but when most English speakers pronounce it (see discussion above), they would actually use [ɹ] as the phonetic realization. You could make up a phonological rule of /r/>[ɹ] as needed, or however you want to explain that in your theory. (Or you could just call the phoneme /ɹ/ which is simpler.)

In the end, there is a lot of variation based on instructors, textbooks, researchers, theories, etc. Just follow the norms from whatever context you're working in-- your instructor/textbook's instructors, or if you're researching look at published papers. It depends on your analysis/theory, but also conventions like whether /r/ should be lazily used for English even though it's not the right symbol phonetically.

The best advice I can give is: (1) follow familiar/appropriate conventions; (2) think about intention (do you want to express an underlying/broad form, or a specific pronunciation, and why?).

---
Note: I had accidentally written [s] so that it made the end of my answer strikethrough above and "[s]" itself invisible (fixed now). My last post might have been confusing, maybe clearer now.

The forum automatically parses [s] as strikethrough unless you somehow avoid that, such as by inserting other tags. Here I'm writing "[[i][/i]s]" to demonstrate (with extra/unused italics just to offset it).

Stress (and other suprasegmentals-- above the level of segmental phonemes) can be phonemic (contrastive: broad) or details of pronunciation (narrow).

So if you are describing the underlying form of the word including stress, then that's fine with slashes.

If you are describing a specific pronunciation, then you would use brackets. Same as for segmental phonemes vs. allophones.

In this case: is there some phonological rule that assigns stress like that, or is it known as part of the underlying lexical entry for the word? I'd assume most theories would consider that to be memorized, part of the word. So slashes.

And the full stop/syllable break used as well as the stress, for consistency? Or doesn't it matter.

It is used when you need to clarify where syllable boundaries are. Usually it's relatively easy to guess just from vowels and typical patterns in the language. So it's not usually specified.

If you're marking stress, then that also serves to mark syllables. But they're two different things, with two different purposes. It depends on your needs.

You're completely correct that usage varies in subtle (and not so subtle) ways. The fundamental idea of broad/narrow, or underlying/final, etc., is relatively clear and seems to be consistent across them, not to say that the implementations are theoretically or technically consistent.

As a case in point, the phoneme /s/ does not have [z] as an allophone, because /z/ is a separate phoneme by most definitions (minimals pairs "zoo, sue"). But there is a regular phonological alternation between s and z, and many people (mistakenly, IMO) sweep those alternations up under the term "allophone".

[z] certainly is a legitimate contextual allophone of /s/. Unless you're disagreeing with the concept of 'allophone' entirely (and we could throw out 'phoneme' as well, but that opens up a whole different set of theoretical issues). You are correct, however, that my phrasing above could have suggested a more flexible relationship between the two in general. Certainly /s/ and /z/ are contrastive phonemes, although both actually can be realized phonetically as [s] or [z], depending on certain patterns/contexts. Yes, there is clearly a difference between "zip" and "sip", but also much more variation in the possible pronunciations (="allophones" in a broad sense) of each.

For students, all that can really be concluded is that these terms are used somewhat intuitively, not always consistently, and in specific ways in particular textbooks or by certain instructors-- so pay attention to the model you're given (examples in class or in the textbook) and then expand from there. Notations and terminology vary, although the general concepts are usually similar. And then you can get into the wildly different interpretations/implementations/explanations in different theories. This is why many textbooks attempt to keep things simple by essentially pretending that theories are "answers" and not controversial, but then sometimes that also ends up confusing the students more because they see alternatives... it's a bit of a mess. But again, just try to follow the examples you have, and don't worry about the bigger picture too much until later. (I'm not discouraging creative thought about these topics at all, but it often won't help you get the "right" answer on a test! Because that "right" answer will vary depending on what your instructor expects. In the most confusing cases, it can actually vary week to week or chapter to chapter in the same class, as the textbook builds up the theory layer by layer-- good in some ways, but confusing if you're out sick for a week!)

I do actually disagree with the claim that "allophone" is a significant linguistic unit, but here I am assuming it and defending it (somewhat).

I have no objection to that. In fact, I think it's very interesting how these traditional ideas break down under pressure from a closer look at unusual empirical data.
However, it does seem useful for the sort of toy grammars taught in classes for analytical purposes. (And that's why the "right" answer can only be defined within such a class...)

In English, s and z are distinct phonemes

Yes. Because they are sometimes contrastive. If you can find a minimal pair, then there is a phonemic contrast. That's the core rule.

whereas k and kʰ are allophones of one phoneme, usually thought to be /k/

Only for notational purposes. There's no reason to write it one way or the other. You could also write a random symbol if you wished, because phonemes are only meaningful in their contrasts. Of course the most natural symbol is the one that takes the fewest steps to actually derive the surface pronunciations. But theories or theoretical approaches do differ in whether the actual "value" of a phoneme is meaningful or not.

the word "allophone" does not mean "sound that another sound can have a derivational relation to" or "has some variation in realization", it is a special kind of derivational relation of the kind "all instances of x derive from y"

OK, I see what you mean now. But I disagree. Allophone means "alternative phone" as in "allomorph" meaning "alternative word-part". An allomorph of the plural "-s" is "-es", and that "same" allomorph is also found for the third-person singular present tense. That coincidence (or maybe not?) shouldn't be enough to decide that the form "-es" is no longer an allomorph of one affix or the other, or both. Uniqueness is not a particularly coherent defining property for allophone either.

the sounds are in complementary distribution, and that is the mantra that millions have memorized ("phoneme, allophone, complementary distribution")

OK, here we go! Yes, good point. But I disagree. Phonemes are unique because they are contrastive. Complementary distribution is not negative evidence for phonemes: they can be neutralized. Only contrastive distribution is evidence. Therefore, complementarity is secondary in a theory. Only contrastiveness has any basic theoretical weight. Yes, allophones would tend to be complementary in an ideal and simple case, but that idea breaks down (or is perhaps incoherent to begin with) in any real language when two phonemes might just happen to have the same pronunciation in some circumstances. If any other phoneme in English, other than /k/ is found to have the alternative/contextual pronunciation [kʰ], then does that suddenly not become an allophone of /k/? It seems to me that almost no allophones could ever exist given the natural variation in pronunciation, especially in fact speech. Examples of where you might encounter [kʰ] not from /k/ would be devoicing of /g/ in some cases, as in /gt/, or maybe the sequence /ŋt/ in certain cases. The problem is that allophones cannot be defined at any particular level of  phonetic variation/derivation, because phonetics doesn't have specific levels like that. Otherwise we could have IPA notations for /broad/ {medium} [narrow]. There is no "medium" level (that isn't even a coherent idea as I understand it), so the fact that you could ever get [kʰ] as the output of any other phoneme would invalidate the uniqueness idea. Complementarity is not a uniqueness relationship, but a failure to falsify contrastiveness. Therefore, either the idea of allophones is incoherent (using your restrictive definition), or we must consider allophones in the more flexible and contextual sense of phonetic variants of underlying phonemes. The idea of required complementarity only works in simplified data sets for textbooks, and relies on an artificial/imaginary "medium" level of transcription.

Another possible meaning for "allophone" would be "sound that is not a phoneme", which could be interpreted differently than either of use has suggested. That would be any segment for which we could never identify a contrastive relationship, such that it must be a contextual variant of some underlying phoneme, but perhaps multiple phonemes. (It also might be the case that this is not a coherent definition because it privileges one phonetic output of phonemes over others that are "just allophones", which doesn't necessarily make sense in all theories and regardless relies on already understanding the system and the underlying values for phonemes, e.g., after a full analysis, not as an observational property of "that's just an allophone" to begin with.)

Personally I prefer the simple "one of several phonetic realizations of a phoneme" because it's coherent and doesn't break down or imagine levels of representation that don't really exist.

"Allophone" can remain a (somewhat) useful concept of just saying 'phonemes vary in pronunciation' and talking about one element of those bundles of pronunciation.

Of course allophones per se are not really a coherent concept at all, in any sense, because we cannot meaningfully define explicit sets of phonetic outputs because they are scalar/variable beyond a simple list of "segments". Only phonemes exist as discrete units*, and allophones are just our "medium" symbols referenced above, although we call that "narrow" and allow various levels of detail.

[*Phonemes may also break down for various reasons, such as archiphonemes, a topic that theoretically interests me, but is beyond the scope of this conversation about definitions of allophone which presupposes phonemes.]