Can Spoils be trashed by Counterfeit?
Spoils is played twice, but is returned to the Spoils pile and not trashed.

Spoils is a treasure providing three coins that reads:


  When you play this, return it to the Spoils pile.


Counterfeit reads:


  When you play this, you may play a Treasure from your hand twice. If you do, trash that Treasure.


According to the Dark Ages rulebook (PDF), page 11:


  If you use Counterfeit to play Spoils twice, you will get [six coins], (in addition to the [one coin] from Counterfeit) and return Spoils to the Spoils pile; you will be unable to trash it.


To answer your second question, Spoils can be trashed, but not when it is played, exactly as you expect.