I can't see any of my sent messages. Is that normal? When I send them, they appear to go out but then nothing is in the sent file and I haven't been getting responses so I wonder if they are actually being sent.
You should be able to see them there, yes. Try sending one to me, so I know if it is indeed being sent out, if you like.
Under My Messages, go to Preferences, then Settings.
Check this box
Save a copy of each personal message in my sent items by default.
Save, and done!