In California the laws regarding security deposits are strict and the California rental law book that I purchased awhile back talks about how to word the lease to tailor it to protect yourself.

She got a cashiers check in the mail and she deposited it.

Also, around here I believe you need a significant deposit and clear proof of funds.

By this point you will have a lot of money tied up in closing costs and also the non-refundable deposit that the investor you are buying it off of has charged you.  

When I get a property under contract is it always necessary to put down an Ernest money deposit?

A mortgage broker working with investors should also have a good network of private lenders s/he can match-up with investor borrowers.While you have what sounds like a good rapport with your existing broker, have a blunt discussion with her to verify she has the experience and reach to embark on this venture with you.

If they are on Supra electronic lockboxes, then you will pay a deposit, plus insurance, and then of course the rental for the key.  

I like to buy them under contract, give the seller a $200 deposit then flip the home to a professional buyer and turn my $200 into $2000 -$5000 immediate cash profit!  

On her lease that I now have a copy of it shows $0 as a deposit while the other unit shows a normal 1st months rent deposit.

As a landlord, you collect deposits to offset some repairs and bifurcate a portion of the rent as reserve over time for larger capital repairs.