Coinbase has to be registered as money transmitter in this case, not you.
They are transmitting the money for you.
Yes. Otherwise, you would have to be registered as an MSB to do any banking at all.
(1) Currency dealer or exchanger.
(2) Check casher.
(3) Issuer of traveler's checks, money orders or stored value.
(4) Seller or redeemer of traveler's checks, money orders or stored value.
(5) Money transmitter.
(6) U.S. Postal Service.
I believe these all refer to businesses or people providing a service, not a person acting on his or her own behalf.