Or, I could try to get one more rent payment (I expect to come this week), then serve them the 30 day notice to vacate by end of June.

Expect the rest to follow.Brad also knows that he’s describing a very different situation than your imagined set up, where you are posting single rooms for AirBnb occupancy inside occupied houses. 

The unit is good condition expect that it is very old and never being updated in the last 20 years.

Check your expectations right at the front; This business is not easy, so do not expect fast and easy success.

By offering only 10-month leases it is our expectation that we will not be able to accept those applicants.  

If you are saying the Net Operating Income (NOI) is going to be $48k ($4k x 12 mo), then maybe you could expect between an 8-10% cap rate, which would value it at $480k - $600k.3.

Some things happen that you do not expect or can control (contrary to the "everything is your fault in life" belief),  but everyone posting hits on this well: Nobody cares whose fault it is or why it happened.

One thing I'm not sure of is location in this case, I suppose I could either look in a well established area and expect equity to increase through remodeling or in a less desirable area that I'm gambling will attract more attention in the future.

You can put whatever limitations you want on the guest that comply with your local laws, but I wouldn't see any major issues, if the tenant thought he was a bad actor they 1. wouldn't want him in their house. and 2. wouldn't have asked you expecting that you would say no.

What is a reasonable amount of time to expect it takes to lease a property priced fairly during the summer?