If they have not yet rebuilt their credit or it is not spotless regarding having zero late payments you reject their application the same as you would any applicant.

Two years ago we offered $310K for a home, they countered back at $330K, angry that they had already lowered the price from $359K.... we rejected it... 6 months later, not sure if it went to foreclosure or private deal, but it was sold for $275K.   

You should have rejected her application when she refused your screening requirements.

I am a fairly new landlord and don't have a lot of experience with tenant interactions. I met a potential tenant yesterday who balked at every requirement (extra rent for his pet, paying for the credit check/backgroun...

Now, what will happen if I reject this guy (assuming he cant meet the requirements)? 

Sometimes lowball offers insult sellers and they will not counter the offer, but simply reject it.

They can accept the offer and you get a good deal or they can reject it and you move on to the next one.If you are trying to do a flip or a wholesale deal then you may need to adjust the numbers but the basics are the same.

I know to get a decent deal I'll have to look at a lot of places, and make a lot of offers that will be rejected.

In the latter ,  you don’t have a choice or your offer will be rejected 

So even though we can go and use fha again as owner occupied we are stuck waiting for at least 6 months because most lenders will not do the loan with fear of it being rejected by the underwriter.