You are limited to 20 assignments normally. However, I've had up to 23 or 24 counting the recruitment actions. The only reason to have more than 150 doffs is just for roleplaying (or too lazy to dismiss your whites).
Yes, Heretic mentioned that he was going to add in the chain progress tracker a few weeks ago. Glad to see it there.
The "overflow" missions are sickbay assignments. I've also had those, up to 22 in my case.