Oftentimes, I see people go into those types of properties with spreadsheet (or worse, their sales agent-assisted) projections, and the projections often do not match up with the reality on the ground. 

A high quality company will be on top of their KPI's and their numbers will match.

So I requested my property manager to forward me all the lease agreement documents and she basically lied and does not match with her company's lease requirements. 

More than the security deposit, and far more than would be recovered by small claims court.

( For example, is it OK if I deposit the cash to my personal checking account and wire it later when purchasing B?)

When do they plan on signing the lease and handing over the deposit. 

Not only is CHA paying above market rents, but as others have said the direct deposit arrives 1st of the month, every month, no hassle.

Wondering if I can be a perfect match at the right price, or if this is just a very bad idea. 

I have been placing offers above ask, and covering closing costs, and earnst money deposit. 

This can mean you'd lose your deposit if not approved for the loan.A buyer's agent in a competitive seller's market will oftentimes recommend:- making a 'best and final' offer since there may not be an opportunity to negotiate with a seller who has multiple offers.