It will do suicidal missions because it obeys your commands. Whether it would be suicidal absent a specific command for example to protect you while you are incapacitated or not commanding a certain action is another story.It's also a fey spirit that knows it can't be killed. It might be willing to do suicidal commands
Maybe
Whether or not it still feels pain is another matter
I personally would rule that the specific verbiage associated with find familiar and commanding the familiar overides even things like suggestion or dominate monster if an adversary were to cast that on a familiar.