The -ing form in English is tricky because historically it comes from two sources (one in adjectival participles ["running man"] was originally -ende, which was contaminated [influenced] by the nominal gerund ["Running is fun"]), and now there are around 4 different uses of the suffix.
1. Deverbal adjectives = participles: "Running man"
2. Deverbal noun = action nominal, gerund: "Running is fun"
3. Deverbal adverb = converb: "Running, I entered the room."
4. Argument verb = infinitive: "I started running."
[You might consider 4 to be an instance of 2, which is fine, but there are also reasons for thinking of it as still fairly verblike, rather than a full nominalization.]
(We can paraphrase (1) as a relative clause; (2) as a nominal clause with "that", maybe roughly "That I can run"; (3) as a subordinate clause; and (4) as a complement clause [not easily in this example, at least in English, though an infinitival paraphrase is easy].)
These can all be considered types of non-finite verbs. And they're all deverbal in the sense that they're derived from verbs but now other parts of speech (with the exception of (4)). Additionally, they behave in some complicated ways in terms of word class (part of speech): internally they are like verbs (they can take arguments: "Running a race is fun.") but externally they behave like other parts of speech in terms of distribution (and a few things like the form of subjects: "His [*he/?him] running was fast.").
There's no real way to tell these forms apart on their own or based on their internal properties. For the most part you'll need to focus on distributional criteria. The most obvious one is substitution: would you substitute a noun, and adjective, an adverb or a verb for the word as it is being used in a particular context?
Typically we don't offer too much homework help here (we certainly won't do it for you) for two reasons: 1) you need the practice; and 2) the particular way your professor expects you to do the problem (the theory and diagnostic tools you've discussed) may be different from how we would do it.
So mostly for reason (2) you should look at what you've talked about in class to see HOW to answer these questions. But I hope that helps some.
As for a more technical discussion of some of these issues, this is a great paper getting into the details of the differences in different non-finite verb forms.
Ylikoski, J. (2003). Defining non-finites: action nominals, converbs, and infinitives. SKY Journal of Linguistics, 16, 185–237.
http://ansatte.uit.no/jussi.ylikoski/filer/defining.pdfRemember: it's very important to distinguish between form and function, especially in English (though many or maybe most languages have some overlap between 2+ of the categories, see the paper above). So when I say that "start running" involves an infinitive, obviously I don't mean a form with "to" as in "start to run". I mean that the function is the same as that infinitival usage. The form is -ing in all of these cases though. We often (over)simplify descriptions of languages by assigning functional names directly to forms (like -ing as a "gerund suffix" or "participle suffix") but that obscures some of the complex mapping from form to function in languages! After all, it isn't the form itself that "means" anything. It's the fact that we use it in a particular grammatical context. And
that means something. Not a problem when a form has just one function (and when nothing else shares it), so it's easy to overlook.