Does anything bad happen if you let the dog pee on you? Is the game still winnable?

Nothing bad happens (except the peeing itself), it won't effect anything else in the game :)!

(*I only know this for sure in the original game, if you're referring to Reloaded then I don't know but I assume they didn't change anything big like that*)

Yes, the dog is just a joke. You can safely ignore him. If the dog annoys you, then you can write "kick dog" or you can try to "pet dog" (these commands give you humorous responses).

All it really does is make someone have a comment since you now smell like dog pee.

You can type in "kick dog" or "pet dog" and in the remakes you can get a kinda lame response if you try peeing on the dog.

Really the entire purpose of the dog is so you don't stand around all night when you need to get laid. Yes, the game has a time limit. You have to get yourself some by 5 A.M. (6 A.M. in the '91 version, there's no time limit in the 2013 remake), and the dog is there so you don't dilly-dally.