So you have to reach the final OMEGA tier to get the MK XII set. But what happens when you are done? Can you rerun everything again in order to get multiple sets to equip all your starships?

Yes. Its not like you have to pick on set and then you're stuck with it for the rest of the character's life.

You'll have to start a new project though to get the items of course. so that means more BNPs and such.

This gooood...

Do you know if the set items are obtained one by one? I don't want to get the whole set as one package, because I still have parts of sets from the time before Season 7.

Is there any chance I can upgrade my rare MK XII Reman set to very rare MK XII Reman set through reputation?

From what they've shown, I believe that once you get to the appropriate tier, you queue up a project to manufacture the item you want, piece by piece. I don't think its an upgrade system though, so if you want the Mk12 VR Reman set, you're gonna have to craft it from scratch.