I'm pretty old-school in my methods (still get actual checks, not direct deposits) and I'm starting to see the need to move into this decade.

How do most property management companies handle security deposits?

Use a phone app and deposit the check.

Always have a paper trail for where your funds come from esp for that 2 month period leading up to your applying for a mortgage.I have a tenant who likes to pay their rent in cash only.This created a problem for me as the deposits were just that cash with no record other than the rent receipt I was unable to claim 800 per month in income for a recent purchase because of this.Now I will only accept a traceable form of payment for rent. 

If they insist on no financing contingency, negotiate down the earnest money deposit to be a sum of money you are willing/able to give up.

Additionally, be sure the prior owner transfers her security deposit to you.  

In other situations where you are skeptical of the seller, you can move forward just by giving the title company your earnest money deposit.

Some attempt to have you open up a joint bank account of their choosing so they can deposit the funds.

I thought of giving her a deposit slip, to make the deposit monthly, as there are bank branches closer to her than our PO box.

I had $1200 security deposit so I was out $300  plus 2.5 months rent.