Should I be able to move the tensioner by pulling and pushing on it?

 

Yes, but how easily it moves helps to tell whether or not it lost it's spring tension. Try to spin the alternator pulley (clockwise) by putting a wrench on the nut.
If the pulley slips easily on the belt, your tensioner isn't doing it's job. If it takes a bit of force to maybe break the belt free, you're ok.

 

Not up and down but towards and away from me. When it's cold its a pain to move tge tensioner but once it heats up there is less than half the resistance

 

Ohhhhhhh, that way! No, it shouldn't have front to back slop. That movement can make the belt to not track straight, and ruin it too. Looks like the tensioner has had better days.