我感冒 (without 了) was rejected. Is 了 always required here?
Here I'm pretty sure the 了 indicates change of state, as in "I wasn't sick before, became sick, and now currently am sick". 了 is very confusing and there's a lot of stuff about it online you can read if you're curious
Yes, it means the action happened
Yes, one of its use is that it is used to indicate a change of state.