Yes he can. Turning the card sideways is only to let you know their once per delve power has been used. The other powers still work.
That's what I was guessing, so I'm hoping I can get a Seth reply or thumbs up in case anyone else goes hunting on the forum.
Yes, the Specialty is "always on" and can be used even if the hero is exhausted
Good gravy that was fast! Thank you very much!