Can any player DH or only those players that had actual MLB DH appearances?

Any player can DH as long as he has starts remaining at some position, regardless of whether he has actually DHed or not.  So if A-Rod has only played shortstop this year, you can still put him in your DH slot as long as he has some shortstop starts remaining.  His appearance as a DH will reduce the number of remaining shortstop starts by one however.