If you sign a player to a $1M, 1 year deal who has less than 2 years ML experience, what is his status the next season? 

Is he a Free Agent?

Automatic contract renewal? If so, at $1M? Or at league minimum?  When do I need to release him if I decide I don't want to pay him?

I believe you have to offer him arbitration or release him. I signed a 69 OVR catcher with 4 yrs ML exp. last season, and hoped to get typeA comp, but I have to offer arb so I'll probably release him. 
Not sure how much he'll ask for, but if you want to release him do it before arbitration hearings.

Well, a player with 4 years ML experience would be arb eligible. I signed bullpen guy  with only about half a season experience, so he'll have about 2 years experience, not enough to be arb eligible.  

It's not the end of the world if I have him in LRB/Mop role again for a $1M, but I'd rather have him at $378K or something like that. I probably have AAA guys who could fill that role for that much next season. 

Okay I see what you're saying. You'll either offer him arbitration or he'll automatically be signed for 378k or whatever. Unless you specifically signed a multi-year deal, I'm pretty sure they can't force you to pay a player more than the league minimum (unless its one of those weird 500k IFAs).

Great. Thanks.