Why not "Ő egy tűzoltó vagy egy katona?"
That's correct as well.
Eehm ... no, it isn't. I mean it's correct, but it duolingo doesn't accept it.
Why is it possible to omit the Ő? Isn't it like "firefighter or soldier", which doesn't refer to him?
"Tűzoltó vagy katona?" is correct, context makes it clear what or whom it refers to. I haven't heard about phrases like this causing misunderstandings.
Do you mean it's clear it's 3rd person?
Why don't you just explain about order in structures? All you guys said was "Order does not matter" and almost all of the time, for what I've seen in the first lessons, it DOES matter!!!!