I had a tenant that was finally evicted, that currently owes me in excess of $40,000.

@Jason Sinclair Seems excessive but also not certain what brand the windows you are looking at, the warranty, construction, tempered / non tempered, double hung vs single hung....  

Either the upper tenant loses their dog, the lower unit tenant can't have a dog at all, or the rear/carriage tenants are affected by the pets on the property.My questions are:1) Can I stand by original lease arrangements, where the upper and rear tenants are allowed a pet, but the lower tenant cannot?

Umbrella is sometimes used interchangeability with excess liability insurance.

I was bummed to lose some of the progress we made, but the sale of the other home paid down all of dumb debt I'd accumulated. 

If an investor PUSHED us to offer it, they'd be waiting so long for us to find a qualified tenant, they'd lose a ton while it was vacant.3) What questions and documentation did you ask for about their PROCESSES before hiring them?

You would lose your primary residence tax benefits but could give you some financing flexibility.

How can I analyze if that will allow me to charge in excess of that amount to the tenants (ex. $+200/ mo rent).

Fortunately, a previous employer called me within two weeks of losing that job and I was back working within a month.

An opportunity cost is the value of the opportunities you are losing by making that choice.