They're leaving today, so once I get down there and check the condition of the house I'll know more.Would you rent to her again (if the house is in great shape)?

The reason that is currently the condition is due to multiple offers and the listing agent/seller being concerned about the appraisal coming in lower than purchase price, which is mitigated by seller asking for no appraisal contingencies and larger amounts of money in the bank. 

Conventional Lender will likely see it is not an arm's length transaction and be more careful to get all the paperwork and not waive any conditions.

For those of you who have considered or actively rent to prior felons, what was your screening process like, and were there any conditional terms you put in place? 

Tenant was inherited to us when we bought the house last year.I want to get last month rent as possible, as I am not sure about the condition of the apartment when he leaves, I may have to use security deposit to cover for damages he might have done to the apartment if there's any.Please share your similar experiences or provide any suggestions that would be helpful.

For considering deals, your buyers are going to want to know stuff like repairs that need to be made, condition of major things such as roof/HVAC/plumbing, property ARV (After Repair Value), a few comparable properties for the ARV you came up with, and plenty of quality pictures of the property.

By improving the condition of the property, you increase the value, which allows you to get more from your refinance. 

I can offer $50,000 over list on day 1 with no conditions to buy a property but I can’t offer $200/mo over asking to rent it?

If you are self managing and the agent is just listing your property for you, you should have a "Listing Agreement" which outlines the terms conditions and compensation you will be paying the agent.

I have an Addendum to my lease that has two conditions that would address this issue (abbreviated): 1.