Is there a -en form for the adjective "trött"?
Adjectives on -tt are the same for en and ett (but gain -a in plural/definite).
Would "tröttnar du?" make sense?
No, that means "Are you getting tired?", usually referring to having had enough of something annoying.
Why doesn't this follow the "V2" rule?
Questions don’t act according to V2, they always begin with the verb. In Swedish grammar analyses, people say that the first slot is ’empty’, so the verb is still in the second place, but that’s a quite abstract view in my opinion.
How about Mår du trött? as a possibility, since we have learned Hur mår du? as an expression? I imagine that I don't quite have an adequate understanding of how and when mår gets used!
You cannot say that