I agree with tonoito. "I want to first sleep" doesn't sound as natural as "I first want to sleep" or "I want to sleep first" (the best answer in my opinion). I'm not saying the answer is wrong, I can think of contexts in which it would be used, but I still think it is the least likely to be heard or used.
While I don't know the names of the types/parts of speech relevant here, the whole/unconjugated verb "schlafen" is used because "will" (first-person conjugation of wollen) comes first in the sentence.
[at least so far] when there's a conjugated verb at the beginning of a simple sentence, the next verb will just be the raw word, regardless of gender/perspective. Schlafen/gehen/ haben vs. schlafe/gehe/habe, for instance.
I think that's the gist of it, but maybe someone else could give a simpler description with better grammar-related vocabulary.
It has nothing to do with grammar. An invented word cannot be correct or incorrect grammatically, it can just not be recognized by any dictionary, or could be too slang-ish for Duo to accept it.
Although I agree with you that it's silly to accept every possible slang version of a sentence because someone is too lazy or too cool to type in the whole thing, saying that it's “not correct” is simply wrong.
I am open to correction from native speakers, but it's because the main verb of the sentence is "will", and the "schlafen" is an extension of it, so is in the infinitive form, not the plural. It's the same in the English - main verb is "want", extension is "to sleep", not just "sleep" or "sleeping".
Idiomatic difference: Technically "möchte" (subjunctive version of mögen) would mean "would like" which in English is often used as a politer form instead of "want" which would be "will". http://german.about.com/library/weekly/aa032299.htm
I'm confused here because "I want to be the first to sleep" which could in the proper context be said as "I want to sleep first" that is if you were speaking to another person who wants to sleep is different than "I want to sleep first" in the sense of before you go to do something else should you be talking to that same person about your own future plans instead... which context is this?