Stake rewards are not added to the admin account or any other account. Therefore, nothing needs to be written to the DB.

POS rewards are simply added to your hot wallet. This is why you have an indication of total sum of user balances and hot wallet balance: so that you know what portion of the hot wallet balance is available to you and what part is for the users. It is also there so that you know what percentage you have offloaded to cold storage.

The cold storage is a security feature, it has nothing to do with how much money the admin account has. You can read more about it in the documentation.

I will look into sending an email notification to the admin when you receive POS rewards.

with regards