In English, I might ask you to "help me [to] do" some task. Would the most (classically) idiomatic Latin equivalent be an ut clause (e.g., "iuva ut faciam ...")? My only reason for hesitancy is that whereas the English suggests collaboration in the "doing," the Latin seems perhaps to say that I (as the grammatical subject of the subordinate clause) am "doing" while you are merely "helping." Am I reading too much into this syntactic difference?
I believe that iuvare ut + subjunctive is not the most idiomatic equivalent. In fact, Lewis & Short do not even mention this construction in their entry for iuvare. You can certainly say it, but I feel the meaning would be more like "assist me so that I may do task X or Y."
No, I think adiuvare in aliqua re or ad aliquam rem is the way to go. In particular, it is often used with gerund(ive)s or other kinds of verbal nouns, e.g.:
Help me lift that box.
Me adiuves ad illam cistam tollendam.
There are numerous ways to use adiuvare, including ut/ne + subjunctive, a plain infinitive and more. But the above seems to me the most common and straightforward translation of help [to] do.