> Does it have a proof of reserves (or proof of liabilities?) scheme to verify the funds of the mint?

Yes, it's already built into the client. For each of your statehain utxos you can click "show info" and it will show you what bitcoin address those coins are located in, which you can verify by looking up that address on your node or on a block explorer. No rehypothecation -- you hold a key to that address, and you hold a "backout transaction" which you can broadcast after a timelock if the mint goes down. (You can see the mint's signature for this transaction in "show info" as well.)

> can this scheme be rolled everyday instead of every 30 days?

You can roll it every second if you want to. You yourself audit your own coins as often as you want.