I Initially offered pretty low, was rejected, and then waited about a month until they countered much closer to my offering price.

If you are sharing your personal space, you have the right to reject people of different races, religions, sex, people with children, etc.

I have an applicant I'm going to reject soon.

A contractor submits their quote and the homeowner either accepts it or rejects it . 

But these are still new and could be rejected by a lender. 

Been rejected, beat by cash and higher offers a bunch of times.

Rejecting on the basis of an ESA dog can subject you to a lawsuit. 

Home insurance won't cover a lot of system failures like electrical.Home warranties are designed to make the warranty provider money; they are slow to respond, quick to reject claims, charge you a trip fee for every visit, use the lowest-cost vendors that provide quick-fix solutions, and are generally not worth the paper they are written on.

Railing and steps which can cause a mortgage rejection on buyers part.

If they reject the offer, then I would play hard ball by enforcing every term of the lease and ensuring they participate in showings or face termination.