How is syntax of polysynthetic languages (e.g. Inuktitut, Mohawk) represented in formal theories of syntax? In many cases, a sentence consists of only one or two words so the syntax tree is rather trivial and doesn't seem very useful. How is the extensive pro-drop dealt with formally?
Polysynthetic languages do almost all the work in the lexicon so a better question would be how they deal with morphology. There's a formal grammar for Georgian and Abkhaz developed by Paul Meurer. He has a lexicon and a nonconfigurational grammar for the NP adjuncts.
It gets more complicated in languages like Greenlandic or Quechua because they can built infinite words (not via recursion though) There's something like a sublexical "microgrammar" that creates "trees" in the lexicon. In other words, syntax is not the problem here, unless the language exhibits incorporation, in which case one has to deal with bracketing paradoxes. I generally like the lexical integrity principle but I think that it simply doesn't work for these languages. This opinion is controversial, though. Quechua, on the other hand, has differential agreement, which means that (lexical) semantics affects morphosyntax. One can vaguely describe the rules but it's very hard to formalize them.
As for pro-drop, the "dropped" arguments are attached in the lexicon and appear at the level of deep syntax, where one can use them to create the logical representation of the processed sentence. But this is not syntax anymore.
The question is too broad to answer completely (to start with, it presupposes a shibboleth to distinguish formal theories of syntax), but the answer is easy for minimalism.
The comment in the question does not apply to minimalist syntax in that minimalist syntax does not assume that the leaves of a syntax tree must necessarily host words. On the other hand, it does argue for common structural features in all languages. So the short answer to "How is syntax of polysynthetic languages represented in minimalism?" is "just like the syntax of non-polysynthetic languages." This area of research has been very active for the last 20 years, so there is no dearth of material to study if you want to know more. I guess the classical reference is The Polysynthesis parameter and I'm personally familiar with the work of M.Barrie but there are dozens of other references.
As Ivan Kapitonov notes in comments, pro-drop is not exclusive to polysynthetic languages. As it is one of the most studied syntactic phenomenon, you'll have no difficulty to find work on this topic either.
Many linguistic theories see syntax and morphology as basically being a single thing: morphosyntax. The same principles and processes occur in both, with the difference between morphemes, clitics and words being hard to define. Their syntax trees therefore include the internal structure of words.