"How can I live without a mobile phone then? Is it inconvenient?"
It may not be convenient but Stallman does it, so it is clearly doable. He gets around the entire planet on his various speaking trips without a cell phone.
"I like using bluetooth earphones or headphones on the go. I am thinking of carrying a computer folded in a backpack, running free software and supplying me with audio via bluetooth. Is there an obvious reason why this would not work?"
I think that would work very well. You may also want to look into https://jmp.chat/ which is run by our very own free software person Denver Gingerich. You get a phone number, and do your calls using XMPP/Jabber.
Denver gave a talk about this at the FSF's yearly LibrePlanet conference this last March. It was called "A fully-free cell phone experience, no baseband required" and can be seen at https://media.libreplanet.org/u/libreplanet/m/a-fully-free-cell-phone-experience-no-baseband-required/
