They're offering the full asking price with a $1k escrow deposit, and they're also requesting $10k for closing costs.

We use Rentec Direct and really love it.It collects our rent, applies late fees, deposits our rents in our bank account, tracks expenses, screens our tenants, markets our properties, generates forms we can send out for docusigning by our tenants, tracks leases, and much more.  

Regarding nonrefundable deposit on lease option to purchase for a home selling for $375K.What does the nonrefundable deposit entail?

You might talk with a real estate/estate attorney and see if it's possible to open a probate, have a Personal Representative appointed and obtain an order from the court authorizing the sale of the property by the PR with the proceeds being deposited into the registry of the court for later distribution when all the heirs are identified. 

We're not sure how to handle the security deposit.

They collect your rent, apply late fees,  deposit your monies into your bank account, market your properties, track expenses, screen your tenants, etc for just a few dollars a month per door.  

So far it has mostly been a deposit of sand.  

I also found out on the day we closed the loan that they have no security deposit either.

You have to deposit cash up front in your bidding account to be able to bid.  

If it gets harder to evict tenants, landlords will require higher deposits and rents to make up for those potential losses.