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

For example, I often see Buyers purchase property thinking there is a $1,000 deposit but then the tenant claims it was $2,000 because they paid the last month's rent.

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?

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

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. 

So far it has mostly been a deposit of sand.  

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

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

The vast majority of lawsuits against Landlords are for wrongful eviction, security deposit disputes, and Fair Housing Violations.