Look up "sequence of tense". It's complicated, but often optional. Here it's either "I forgot I had" because the whole thing is in the past, or "I forgot I have" with reference to the future.

Or you could say "I forgot I will have", which sounds a little awkward/lengthy, just like "I will have...", as if you don't have it scheduled yet or something. The meaning of "have" is a relevant consideration in general: it is a stative verb, and it is true not just at the time of the event, but also in a sense when it is planned, because you "have" it then, as a plan. So "I had it" might work in the original sentence for this reason too, because it already was a plan, even though the actual event has not occurred yet. It might be clearer if you used a different verb, but "have" is interesting.

Yes, that is part of the explanation (with some other complex details).

It is not grammatical to say:
"I forgot I have an appointment yesterday."

(The only way that would ever be uttered, still awkward, would be if somehow having an appointment yesterday were still ongoing, as in maybe a weekly repeating appointment with the same person where "yesterday" really means "on Mondays" or something like that, but coercing the sentence to work like that would require a very specific context. Assuming it's a one-time appointment, that just doesn't make sense, unless somehow it was a multi-day appointment that the person was late for, and still planning to go now, or something like that. Saying "I forgot I have an appointment scheduled for yesterday" could make slightly more sense in that context in the sense of reading notes on a calendar and saying "oh, I see what I have written there, I have an appointment yesterday", referring to the location on the calendar, rather than the time, but disconnected from the event itself, and almost certainly a missed appointment, or maybe one the speaker had forgotten about, if they are being questioned about their activities the previous day, and are not even recounting what happened as a story "I had an appointment, and then", but actually listing out writing on the calendar, or maybe if someone else was reading out loud from it, like "OK, the murder victim has an appointment yesterday, a vacation tomorrow, and...")

Yes, the present tense is allowed as long as you're referring to something ongoing. Note that this might not be properly categorized as the "present" tense anyway, but more of a habitual description ("I eat pizza [sometimes]" vs. "I am eating pizza [right now]").

1. By saying "ongoing", you mean something that is still true or going to happen?

Yes/maybe. The details are very complicated, and depend on the lexical aspect of the verb, among other things.

It's better to think of this in terms of usage choices than grammar rules: when speakers choose to use words in a certain way, they are presenting the information in the sentence according to that grammatical model. If someone says "I forgot [PRESENT]" then they are indicating that the "[PRESENT]" part of the sentence is still ongoing or relevant in some way. (That's why you can "bend" the rules in context, by the way.)

2. If something is "ongoing", does it make much of a (stylistic?) diffeence if we change the verb form (e.g. from "have" to "had" - as in the already mentioned example) or not, saying everything in the past tense?

Very little difference. Aside from substitutability (that is, if you were talking about a different situation, whether you could use either of both of those same forms then), the two forms are basically equivalent in the contexts where they both could be used. But two forms are never equivalent, because they are associated for speakers with other situations in which they might be used, or even just the way other speakers have used them (in specific cases) in the past. Here there might be a very slight sense of formality with the past tense (just like other cases where using a marked/past form is a bit more formal), but there's no substantial different meaning to my ears. It does, however, give a suggestion that the speaker is or is not thinking of present-relevance.

In short, instead of asking what the rules are, try to understand why speakers would make the choices that they do. In this case, it's simple: they must choose one form (aside from some kind of awkward circumlocution like "had and still have"!), and therefore they choose either to keep it in the past tense (the "simplest" in a way), but removed from present-relevance, or to highlight the present-relevance by using the past tense. I suppose it might be implied that using the past tense is instead of indicating that the event is ongoing, so you'd want to use the present if possible, to avoid that implicature, but I don't get a strong intuition like that with these sentences.

To make things more confusing, you could also say:
"I forgot (that) I was going to have..."
(But that might suggest it did not or will not occur. Future-of-the-past has its own complications.)
Or instead of the present (with essentially the same meaning) you could also say:
"I forgot (that) I will have..."

Honestly, I'd guess that these forms might even be mis-used by speakers (careless usage that doesn't literally line up with what they meant) just about as often as they are used with any specific intention in mind to highlight relative timing of events. English is fairly loose about permitting this sort of thing (here, being able to mix past, present or future forms all in the same sentence!). The decision might simply be whatever seems most natural, i.e., something like what they heard most recently in their input. But sometimes there is a slight distinction, e.g., to highlight present-relevance.

I don't have a more explicit answer for you about this.

Tell me please if the meaning of the following question is the same with either verb form. ...
At what moment in your life did you realise that goldsmithing was/is actually something that you wanted/want to do?

Yes, the meaning is functionally equivalent in that context.

The minor distinction would be that past tense would be more appropriate for interviewing someone about their biography/history (asking about the past), and the present tense would be more relevant for interviewing someone about their current job (such as career advice for others who want to do that).

The details are extremely complicated, and I don't know that I can explain them all here (or honestly that I completely understand all of my intuitions about these forms to list out every contingency).

As I've said, sometimes these are essentially interchangeable, as in your first example: "I didn't you know you live(d) in a houseboat".

In the second case, it would actually not be acceptable to say "I thought you are still on Holiday", because that would be counter-factual. Literally I suppose you could say "I thought you are", but that seems really weird to me, because there is some implicit negation here indicating a previous false assumption, that has now been corrected. The discourse function of "I thought [false statement]" is to emphasize the correlated idea "Now I know that actually [true statement]".

"I thought you work from home” (not "worked").

I'd prefer the past tense. But "work" (here) is better than "are still on holiday" (above) because it's not such a bad assumption: it's still possible that you usually work from home (sometimes, but not today), but it's not possible that you are currently on vacation while I am standing in front of you.

Given that, is it actually common in everyday speech to use the present tense after "I didn’t know…" or "I thought…" constructions?

No, it's not common with "I thought" because that is correcting a false statement.

With "I didn't know", that's different, because it's stating a true fact now, so the present tense might work. The complexity is that "didn't know" is in the past, so it would need to be a long ongoing event for it to work with present, or especially a habitual usage, something like "I didn't know you like pizza!", or maybe "I didn't know you are still sick". That works. But I'd prefer the past tense for both of those. (To my ears, the present is OK with habitual usage like "like pizza", but awkward with a real present tense like "are still sick"-- you can't have known that earlier, because it might have changed in the time since you "knew". But people might say that, just awkward.)

--

For all practical purposes, this is essentially irrelevant. You'll be understood well. You might not understand some subtle nuances (like whether the speaker intends to express present-relevance), but that's rarely an important part of a conversation, and often redundant in context anyway.