How do finite verbs work in questions?

In order to understand what Flesch is trying to say, you should first of all realize that grammar isn't his focus, or even interest. For example, on pp. 71-72, he rewrites

[1] American belles-lettres also give a much more faithful and adequate picture of the entire civilization to which they belong than literature abroad.

as

[2] American belles-lettres picture the entire civilization to which they belong more faithfully and adequately.

Now note that the distinction between [1] and [2] is not one of passive vs active voice, nor one of finite vs non-finite verbs. In both, the verbs are finite and in the active voice. Instead, it is about choosing to use picture as a verb rather than as a noun.

Consider next the following segment from the book (pp. 66-67):

The point of all this is, of course, that I am talking here only of those words that are used as verbs in a sentence. They are what the grammarians call the "finite active verb forms" and they are the only ones that have life in them. Hearing of verbs, you probably think of passive participles and infinitives and gerunds and all the other fancy varieties that have plagued your grammar-school days. Well, forget about them: for all practical purposes they are not verbs, but nouns or adjectives—lifeless words that won't make your sentences move. The verbs want to use are those that are in active business doing verb work; if you use a verb in the passive voice or make a participle or noun out of it, you have lost the most valuable part in the process: it's like cooking vegetables and throwing away the water with all the vitamins in it.

Note that although he used some grammatical terms, he immediately told you to forget about them! Which is just as well: as John Lawler implied in the comments, Flesch didn't actually seem to understand grammar all that well.

So one should not get hung up on non-fnite verb forms. In particular, I would say that Flesch is not advocating a ban (or, rather, avoidance) of e.g. the continuous tenses. The sentence

[3] I am running.

is as 'active' (according to—I would say—Flesch's definition) as it can be, despite the fact that it contains the present participle running (which is needed to form the present continuous tense).

So while knowledge of grammatical terms is helpful when trying to implement Flesch's recommendations, these recommendations are not reducible to the avoidance of certain inflectional verb forms.

Questions

Now let's consider your sentence

[4] Is your biology keeping you from developing?

(One slight correction to what you said in your post: keeping isn't an infinitive, but a present participle.)

The 'canonical' counterpart to [4] is

[5] Your biology is keeping you from developing.

That is an active-voice sentence, in the present continuous tense. In both cases, the active verb is the auxiliary is, while the complement keeping is a non-finite verb form. But Flesch would not condemn either [5] or [4]. They are both 'active' in his sense of that word.

It is a matter of English grammar that all closed interrogative clauses (i.e. those that invite a yes or no answer) must feature subject-auxiliary inversion; if there is no auxiliary verb, then do is used (that's called 'do-support'). In both cases, the lexical verb appears in a non-finite construction, even if it was finite in the canonical (i.e. declarative) counterpart. So, for example, I kept it hidden becomes Did I keep it hidden?

But, again, this sort of non-finiteness is not what Flesch is objecting to.