This allows us to collect a non-refundable down payment that goes toward the purchase of the property, which can be used to cover any vacancy costs if they decided to no longer lease the property (and also not purchase the property). 

Rent collection issues can be mitigated by programs such as Section 8 (aka rent assistance) or VA housing, where rent is guaranteed by government.

We collect the money then take our cut and pass the funds up the food chain to my Capital company and close on the notes splitting them to our buyers.

Once you finance a deal, you hire a servicing company to collect payments and deal with collections and foreclosures.If you are putting in work in order to get money out of your real estate (fix and flip or buy and hold), then you are self-employed. 

Abilty to Repay Rule says:The lender must determine your “ability to repay.”Before you get a mortgage loan, the lender generally has to determine you have the ability to repay the loan.The lender must collect and verify your financial information.When you apply for a mortgage loan, you have to give the lender certain financial information.The lender has to check the information using reliable documents, such as a W-2 or pay stub.

No utility bills in collection.  

If they refuse to let the seller pay off the mortgage, the seller can simply sue the lender.And it'd be real easy to collect.

If that's the case, then it would depend on the specific laws in the state concerning child support collections. 

Is there any type of paperwork required (i.e. registering my business license in the Inland Empire since it's a different county) in order to start collecting rental income? 

A local attorney can tell you if it's worth chasing the seller and don't forget, anytime you sue you pay, then only if you win, you get to try to collect. :)