Despite all this, the seller is not willing to release the EMD, he offered 25% of the $5,000 EMD which is insane considering that we had a financing contingency in the contract.

It seems to offer both conventional loans as well as DSCR loans which you can get based on the income of the property.

If I don't sell the rental, what kind of lenders offer loans so I can buy back the other portion that was gifted to the nonprofit?  

The seller may be offering to sell the property on a contract for deed, rather than a transfer of deed.  

They sold right away with multiple offers.

Owners mistakenly ASSUME all PMCs offer the exact SAME SERVICES and PERFORM those services EXACTLY THE SAME WAY, so price is the only differentiator – so, they often select the first PMC they call or that calls them back!

If a buyer is offered a 2-1 buydown (starting at 5.25%, increasing to 6.25% and then to 7.25%) with all closing costs covered except for $3,000 (to be paid by the buyer) on new construction through the seller's preferred lender, and signs the purchase contract in April 2024, the house will be completed by the end of September 2024.

Plus, the emotional peace of mind from owning a home outright is significant, offering both psychological and financial benefits.

Depends on your local rules and regulations, but in the state of Virginia such a practice is perfectly lawful.PS: Do not take model legal language and incorporate it into your contract unless such language comes from an attorney. 

Due diligence and could have done better research before putting offers in. in second place would be turnover