First time buyer - Inheriting tenants/raising rent - need advice

[Jason Hines]

We are set to close on our first duplex next Monday in Pennsylvania. We will be living in 1 unit and would like to continue renting to the existing tenants in the other unit. They have been there 10 years, have taken excellent care of the unit, and were very nice when we met them during the showing and inspection.

My questions are in regards to taking over the tenants, timeline for certain things, and general tips. 

Their lease is set to end at the end of October (currently month to month) and the seller will be giving us the prorated rent for October and will transfer the security deposit to us as well. Now this is where it gets fuzzy for me. The unit should rent for around $1250/month but they are only currently paying $800/month. We would like to raise it $1200/month at least which I will assume make them unhappy and possibly unable to pay.

We plan to make them reapply to rent the unit and do a full background check, which should allow us to determine if they can afford $1200/month. If we determine they can't afford it, how quickly can we force them out? Since we're new owners and since they're not under a lease as of 11/1, can we legally make them leave on 11/1? Now, I'm not a jerk and I realize that 3 weeks notice to move out is rough, especially in today's current world state and the holidays upcoming. We've budgeted for and are OK with letting them pay current rent, or $1000/month through the end of the year so they can find a new place. If we do that, what specific verbiage should I use in the lease to make sure I can get them to leave in Jan or Feb? If that's a bad idea, please tell me. Any advice you have would be greatly appreciated. I truly don't know.

Now, if we determine they can afford it, but they refuse to pay a 50% increase, can we force them out on 11/1? 

And finally, if we determine they can afford it and they agree to pay the increase, should we have the new lease start right on 11/1? Should I also collect the extra money for the security deposit at that time? I feel like the answer is yes to both of these questions. This seems the easiest for everyone, but might not happen.