My tenant, who received the company check, addressed to our company, was able to cash/deposit it at their big name bank.

When we moved out, the landlord insisted on keeping the entire security deposit of $1,000. 

However, given our current state I'm preparing to set-up my own website and/or post on various channels that don't appear to offer the host insurance and I know asking the Guest for security deposit or to run background checks can be a deterrent for short stays.

As @Joe Gore  stated I've always believed as well that as long as you have a secured interest in the property (i.e. a signed purchase contract AND a Earnest Money Deposit) then it is completely assignable as all contracts are unless stated otherwise in the contract (A HUD 5.0 says you cannot which is why I never use one in my transactions).

Deposit or invest in $100,000 increments CDs, money market, etc at that bank.  

Deposits it may put you at risk but you have to judge based on your screening.

Should I ask for a deposit in their daughter's US account?

I would like to move out of the apartment that I am in, and I will need deposit-cash for a home in Odessa.

Require a deposit that's at least the cost of a months worth of rent. 

If you do, I would charge a pet security deposit, a pet fee for processing paperwork (require vet shot records), and monthly pet rent.