Well, do you *need* another credit card? If your Amex and Visa's are enough for you, you might want to reconsider opening new credit. But you are an adult and know what you are doing so to answer your questions:
I would recommend Chase Freedom (but you would need to apply by phone to request MasterCard or find the application from MasterCard (under find a card) directly as Chase primarily offers and markets only the Visa online on the Chase website).
1. Chase Freedom has no annual fee
2. Actually, it technically is a *points* card. They are called Ultimate Rewards, but they can be redeemed for cash 1:1
3. This I am not so sure on. They pulled Experian for me in Arizona. Check out Who Gave Me Credit
to see which lenders pull Equifax.
4. It has a pretty sharp design. Cool metallic blue with "Freedom" and "Chase" using hologram material (that incidently gets destroyed when you use the card regularly). Visa Signature is even cooler as the Visa logo and "Signature" also use the holograph material instead of the white border around the Visa logo. As far as I know, there is no World Elite Freedom so the MasterCard logo will be orange and yellow.
5.Chase is, I believe, the largest credit card issuer in the US. If not #1, they would be #2 behind BofA.
Chase, like Citibank and Discover offer quarterly 5% bonus cashback offers which can add up quickly. Chase usually also offers a sign up bonus and if you have Chase checking, you get an additional 10% cash back just for having checking.
Other than Chase Freedom, the only other cash back MasterCard that I can think of is Citibank Dividend (however, Citibank now only markets the Visa) and that lame Capital One Cash Rewards card. I would avoid that unless you want a $3,000 credit line, all three bureaus pulled and only 1.5% cash back (1% plus a .5% annual bonus).