It's bad grammar. 'He/she' is singular, 'they' is plural. On the other hand, I consider it a significantly lesser transgression than just introducing new words (with poor aesthetics, IMO).
No, it's not. It was made-up in the 1740s, and has never really been followed as a rule that closely.[1]
Or from Geoff Pullum[2]:
>"Avoid singular they if you want to; nobody is making you use it. But don't ever think that it is new (it goes back to early English centuries ago), or that it is illogical (there is no logical conflict between being syntactically singular and semantically plural), or that it is ungrammatical (it is used by the finest writers who ever used English, writers who uncontroversially knew what they were doing)."
Or from Geoff Pullum[2]:
>"Avoid singular they if you want to; nobody is making you use it. But don't ever think that it is new (it goes back to early English centuries ago), or that it is illogical (there is no logical conflict between being syntactically singular and semantically plural), or that it is ungrammatical (it is used by the finest writers who ever used English, writers who uncontroversially knew what they were doing)."
1: http://www.newrepublic.com/article/112896/tyranny-pronouns-f...
2: http://languagelog.ldc.upenn.edu/nll/?p=89