This FCC approval is to be able to talk between orbit and the ground in the US. SpaceX needs approval from other countries for the same, in order to offer service there.

Launch approvals are a separate thing.

Per the Outer Space Treaty, no. Under it, space is free for the use and exploration of all countries. The US is liable for SpaceX in-space fuckups, though - if they crash a satellite into something, the US is responsible.

Good question. I doubt the Russians or Chinese consult the FCC for launches.