Ugh. There are much better solutions than keeping user funds in the hot wallet (fully cold storage with manual withdrawals, multi-signature wallets), but many "reputable" businesses STILL uses them. I don't understand why. If you want to store your customers funds online, do it the right way, or don't do it at all.
You cannot automate a cold-wallet scheme. Any automated website or tool will require a "hot wallet" of some kind. The more funds in the hot wallet, the longer the website / BTC Bank can go automatically. Customers like having funds available to them immediately... among other things.
Yes, you can. You can have the hot wallet only deal with multi-signature outputs, and have these approved by separately locked down servers running behind TOR, for example, using out-of-band mechanisms for approving transactions.