However in other areas (like FL) it is very specific in the LL/tenant law that partial payments do not stop any eviction proceedings unless the deficiency is fully cured. 

How I proceed from there depends if this request comes prior to the tenancy beginning, or after we have already signed a rental agreement.

Judging by how far off your offer and inspection report are from their BPO and counteroffer, you will know more or less how to proceed at that point.Make sure your files contain the following always:1)MLS listing showing you have had the property on the market for a while and haven't been able to sell it at the demanded figure, so therefore you need approval on a lower offer to make it work.2) Inspection report documenting everything that needs to done to the place to bring it up habitable standards.

If rent is not paid in full by the 6th day, eviction proceedings will begin.Payments received by the landlord when there are arrearages, shall be credited first, meaning any outstanding balance, and then applied to the current amount due."

If they do not pay I proceed to eviction.

If we proceed this route, I may try to open up the space more between living and dining with french doors or similar.That said, I don't really like entering sideways at the front porch.  

If you replace the property with similar property within the alloted time, you won't have to recognize the gain until the sale of the asset later down the road.Should you choose not to replace the property and keep insurance proceeds, the gain, or loss(if any) will be taxable income during the year it was realized.

And once the rent is paid, you won't be able to proceed with eviction.c) You need to get a copy of the lease to see if there are any other breach.With all that said, if there is a bonafide "arms length" lease in place, you will have to honor the lease, unless they breach some terms.

If there is no response then the foreclosure will proceed. 

And you must use all of the proceeds in the next purchase.Of course within these 6 requirements there are all kinds of little quirks and nuance that your QI guides you through.