This is correct. Fastmail have actually spoken about this, and their silence on any particular topic can possibly be seen as a canary. They're a great company trying to survive in a hostile environment. I really hope that the new Australian federal government will get around to winding back some of the draconian laws the previous mob brought in, but I'm not holding my breath.
It is not correct. In order to need a backdoor, the service would have to be E2E in the first place, which it is not. Uninformed HNers bring this up on every Fastmail thread like it's a big deal when it has zero actual impact on Fastmail at all.
E2E isn't a requirement for something to be called a backdoor, even though governments are constantly demanding E2E backdoors.
An attacker having shell access or a government getting plaintext dumps of whatever email conversations they want (when users don't expect it) are perfect examples of backdoors. AT&T giving the NSA a secret room for them to suck up all comms, encrypted are not, is also a classic example.