Would disagree here. Why would you pay for banking services when the bank EARNS money with the money in your bank account and with transaction charges (you don't always see the charges, but they are there!)?

The bank should be paying you for being willing to give them your money for them to care for. Without you money, they can't make money (give interest paying loans for example).

The bank should be paying you for favoring transactions with them rather then with other banks. Without your transactions, they're just a safe deposit box (or a mattress below which you can store your money if you will).

Seriously, banks should be paying customers to thank them for their custom, as some AAA+ banks in Europe do and have done for years.

Managed online, yes
completely free, no. All is free, but you have a transaction fee when withdrawing money from an ATM, and obviously when you're overdraft.

are those the only two fees?
how much is the ATM card?

As far as I remember, they have a form online - when you fill it in they send you a complete info pack. Would suggest you do that and read their T&C. Or go to your closest Axa branch and ask.

From reading this thread and others, I'm assuming these options are all banks? Is there a financial institution similar to a U.S. credit union in CH?