They say that if I don't respond, they will automatically pay me and proceed with the necessary construction. 

I’d rather take a property down the street where I can tweak the revenue and expense levers than an out of state property where I need to trust that the property manager is putting in good tenants and I am not subject to getting oversized “automatically authorized” charges like replacing a $100 toilet for $500.

Based on what you've told us, I would deny them automatically.

I don’t know whether a background/credit check is automatically rolled into the ‘application fee’ or not.

I too used cozy and all of my properties/rent collection was automatically transferred over to apartments.com.

Two years ago just giving it your best shot on theming meant automatic success. 

Does the contract automatically become null and void or does it give you, the seller, the Right to terminate?

Automatic.

Her complete mortgage is paid each month from the AirBnB income, and she rarely even sees her guests because they have automatic checkin to the suite and no access to rest of the house.

They seem to still offer the service you are looking for Pay Rent Online - Automatic Rent Collection for Landlords | Apartments.com