Most dogs always are, in a sense, "hungry". They will go for any scrap of anything they can get. My 2 year old BCx is an exception to that, although my 16 week old BC pup acts like she is always hungry, too. So don't worry! Your pup is not starving.

Puppy food is fine, it won't do any harm to your older dog although it could make him/her gain weight because puppy has more calories and proteins meant for growing puppies. He/she may also go #2 more often than when he/she was on adult food.