I'd immediately go to a higher deposit, several months paid up front, etc. 

My realtor/PM says $0.95 to $1.05 per sq. ft. is a reasonable monthly rent for a 4 BR/2 BA property in that area, which (according to the BiggerPockets calculator estimate I did) is enough to cover the monthly payment plus the above costs I mentioned.I anticipate putting down my deposit and earnest money in the next few days. 

The seller could simply retain the deposit.

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

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

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're not sure how to handle the security deposit.

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.  

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. 

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