It's actually a bit more complicated, as Swisscard isn't a bank per se: It's a joint venture between AMEX and Credit Suisse that issues a variety of credit cards including all AMEX, Visa and Mastercard.
They do that not just independently but also for other banks, however the cards offered on the Swiss american express website are issued by them, but apart from that they are very similar to AMEX cards issued anywhere else.
Fun fact: Just to add confusion, there is actually a slightly different an AMEX Platinum issued directly by Credit Suisse, which has slightly different conditions but looks exactly the same. It will be interesting to see what happens with the merger of CS into UBS.

To answer your questions:
- Yes, they do earn membership rewards points!
- I don't know. This is a standard world mastercard, that isn't officially listed anywhere, however given some acceptance issues, you can get it with most AMEX cards offered by Swisscard upon request.
- Yes, Membership Rewards points can be transferred to partners.