Banks for savings or "stashing cash"

Any bank which is safe will ask questions about source of funds and in fact a bank that do not are placing your money at risk as it is only a matter of time before they experience regulatory issues. Likewise any bank in SEPA will make enquiries about source of funds and share information about you, so I think you should rule out SEPA or EU locations.

The banking location will generally not impact on the tax liabilities that exist. This will depend on things like company location, control of the company location and their respective tax regimes, but in terms of countries where the company will not incur tax you can look at places like Seychelles which will cover points 2 and 3. Banking for a Seychelles company where you will be asked questions about source of funds (and you really have to accept this) I would suggest looking at Mauritius, Puerto Rico, Cayman etc which will allow for remote opening, no maximum deposit limits, limited due diligence albeit source of funds will be checked and will accept transfers from EMI's.

Another method would be to find a Trustee to take care and manage a trust account for you. This is as far as I know still 100% anonymous and no one will ever be able to get information about you and the account.

It's an expensive way to maintain banking but it's still an option.

Otherwise you may want to stash your money in Gold somewhere since you can buy Gold anonymous.

The only issue with using a Trustee to bank is that all beneficiaries will need to be identified. If you are not named then you have no rights and as the bank will know the account is being operated by a Trustee they will expect beneficiary information and ID to be available.