If you abandon a match then competitive will be unavailable for a period of time. The first abandon has a 30-minute cool down, the second is 2 hours, then 24 hours, and one week. If you havent abandoned for a week you would cool down by one level (for example, from 24 hours back to 2 hours). I don't know if TKs are handled differently than kicks, but I don't think Valve have specified how many kicks you can have before the system slaps you with a cooldown. I think after too many kicks you get a week unlike abandons where the penalty increases with frequency.
I think TKs are handled the same way that abandons are handled. I never got a cooldown for teamkilling - but I was on a 2h one for abandoning the day before. So I guess that explains why I got a 24h one. I'm now a "third level offender" in Valve speak. (1. 30 min, 2. 2hr, 3. 24hr 4. 7days)