Is there some plans for special badges for chat?
There is now the Talkative bronze badge, mostly to publicize chat per-site. See the badge page for more details.
I hope not. Keep the third place out of the rep race IMHO. Incentivizing it just seems like a good way to shift the focus off of what it's actually for, and might bring people to it for the wrong reasons.
How about adding just one easy bronze badge for chat? It draws some attention to existence of chat feature but doesn't require anything massive to achieve.
[Chatter] - Posted at least one message to chat
We don't want to encourage discussions too much, I believe. After all, the core of SOFU is Q&A, and too much discussion in a separate site can scatter knowledge beyond the ideal point.
Giving badges for using the chat would encourage people that don't want to use the chat to use it, resulting in people that don't give a flying cow about the discussions at hand, resulting in a far lower S/N ratio.