They say that the temp only goes to 60 degrees and that it needs to go to 40 degrees. 

You never know who will apply, and you are not obligated to accept them.

This does not mean you can leave the unit vacant for 6 months (not that I'm accusing you of wanting that) but it also does not mean you can be forced to accept a less desirable tenant nor can you be forced to lower your rent.

However, $1K is much higher than what we pay.We do not allow partial payments and I find it curious that your PM accepted that.  

However, this price is much lower than the listing, and the seller is unlikely to accept it.

If someone had an "event" in their life (illness, divorce, etc) then we may be willing to accept that, but if someone has had their vehicle repoed, they have items in collections, and a history of late payments, we'll pass.

While I never before or since accepted any like offer, the amount I was paid for minimal time was too good to pass up.

You can use a  hypothetical  purchase price and loan amount for the pre approval process  as this will get revised to the actual figures once you have located a  proeprty and have an offer accepted

After much back and forth with our attorney and their regulators, etc they decided series LLC’s should be a part of their customer base and have rewritten their policies to accept them (and us).

A concession is primarily for the buyer(s) benefit, it can help seller(s) to sell a property but there is really no advantage to a seller accepting an offer with a concession except that it is incentive for the buyer(s) to focus on that property, it makes the terms more attractive.