Yes, Visa, MC and Amex take a cut. This infuriated shopowners to no end when they raised their tariffs, prompting a wave of "no charges under 50kr" or "3kr extra if you're paying with card (under x kr)"
No, the banks take a cut. The interchange fee that goes to the networks (VISA & MC) is very low (don't know the exact amount).
AMEX is another beast. AMEX is not sold through banks and AMEX can therefore set their own very high rates. This has caused many retailers to stop accepting AMEX altogether. IKEA being a good example of such a retailer.
AMEX is another beast. AMEX is not sold through banks and AMEX can therefore set their own very high rates. This has caused many retailers to stop accepting AMEX altogether. IKEA being a good example of such a retailer.