I would want the 10% to cover a lot of that stuff: they're charging for a lease renewal?

The judgement is good for 10 yrs and can be renewed for another 10.

I wouldn't consider having anything to do with elder care except as a triple-net deal where I owned the property and rented on a long-term, renewable lease to a very experienced, reputable company with multiple other locations I could check out first. 

Question is....should I not renew their lease (month to month) right out of the gate or let him stay? 

If you are negotiating 5-10 year leases, many tenants are looking for renewal options as well which could take you out 20+ years. 

We have also recently learned (the hard way) that we should start considering doing background checks on "inherited" tenants through a property purchase before renewing their leases. 

Talk to several companies and ask about their policies on late rent, maintenance, eviciton, how quickly you get paid, cost to renew, cost of new tenant, add on fees for maintenance?

Learned how to not renew a lease and not return a deposit.

You will have more closing cost up frontOn the fixed however by the time the bank charges you every time they renew the rate and loan and get closing cost it’s about same

Make a rent increase schedule that shows the increment prior to lease renewal time, just make sure that it falls within rent regulations.