Can one daven Drunk and what is the Level of drunkenness that disallows one from Davening?
Preferably one should not do so, though after the fact his tefillah is valid - unless one is so drunk that "he would be unable to talk before a king," and then his tefillah in such a state is invalid. (Shulchan Aruch, Orach Chaim 99:1)
Rema (ibid. :3) comments that "we're not concerned about slight drunkenness" when it comes to present-day wines (which are weaker, either in alcohol content or in viscosity, than Talmudic-era ones) - the more so if one is davening from a siddur. Nevertheless, Pri Megadim says that it's not a proper thing to do.
If someone is drunk enough that they wouldn't meet with the President, they shouldn't daven. (Shulchan Aruch talks about "drunk enough to not be able to stand before a king", but I think it gets the idea across.)