Is "tweet" a proper verb now?
Why not?
Tweet was already a verb meaning "to chirp", and many people are using the secondary definition of "posting to Twitter". Is there a good reason to not consider it a valid usage considering its adoptance by a great amount of people? English is an evolving language, after all.
I don't see why not. As another example, lots of people have been using the word "text" for years (meaning, of course, "to send a text message.")