The seller just had a baby and they purchasing a home that is contingent on the sale of their co-op unit..

Just have your seller (the owner of the property) sign a lease option agreement with an assignment contingency.

If it is very large you likely need to contest the EM be returned to you.Did you not give cancellation notice based on a finance contingency in the contract?

As a matter of fact, I usually have to waive title and inspection contingencies as well to have a fair fighting chance.4) I usually get 3-weeks to close.

It is considered a contingent liability on your personal balance sheet when you go to fill out a PFS or personal financial statement.  

They will escape with a contingency whenever necessary. 

Do I take the issue up during contingency, or is creating these new rules a right I have as the owner, provided that I follow the lease? 

This is simple and is permitted by right.Sometimes a city will require that the current titleholder make this request, so you may have to get the other seller to sign.Also, make your purchase offer contingent on approval of the multi-family use.

In order for them to evaluate the property along with your financials, you would also need to have a sales contract for that property, typically with a financing contingency.  

Any offer you put it make it contingent upon and inspection and get a structural engineer review it (not a home inspector) - note have a home inspector review the rest of the house as well