But I don't want it to look like I'm committing any sort of fraud by claiming a laptop or something.

This guy is an absolute fraud and I will be seeking legal counsel against him. 

Or maybe a tenant prefers a dark middle unit so that's the one that's higher.In another case, I have one building with almost identical units and the same lease start date, but the rents are different because one tenant has a Section 8 voucher, which came in slightly lower than the asking rent.

Answer: Yes.Assuming the parties have nothing in writing, the seller is likely able to do this based on a legal concept called The Statute of Frauds.

One argument I think we should go deeper into is the straw purchase and mortgage fraud because many of the examples I have seen would not be considered mortgage fraud unless the person is, in fact and proven, to be doing so to hide the transaction. 

The best frauds have an element of truth or they wouldn't work.

Send tenant 2 identical letters demanding payment - 1 certified, 1 regular mail with 'Address Correction Requested.'.

Subject-to sounds an awful lot like mortgage fraud to me.

When you get credit/background reports back, you have to verify your identity with Experian. 

A personal identity is stolen every 2 seconds.It is good you asked first.Do you have an accepted offer and need the funding to close?