To bake the active action only; under Bake Animation I uncheck “NLA Strips” (unless of course you’re using the NLA strips) and “All Actions”

As for removing an action. If it’s not going away after hitting the X, it probably has users associated to that action yet.

Which means its probably still hanging out in the NLA editor.

Delete it here and when you go back to the action editor you should see that there are 0 users on the action. Save, and the next time you open the blend file it will be gone.

I hope that helps you out.