Can you even make a universe in a single player session or will you just live forever in the medium since you gain unconditional immortality?

You cannot create a Universe in a Dead Session. We also don’t quite know if EVERY Dead Session grants you Inconditional Immortality if you win. I think it does, though, merely because I think Dead Sessions are a Trap.

Like, they’re a Punishment for people starting such a Session. We know that much. They’re made basically unwinnable, and Caliborn only managed because he was a Lord of Time AND also had extreme luck during the First Break, giving him an edge with three less planets to take care of. Otherwise even he would have failed, arguably. Now, however, the thing is, once you beat the Session… What do you do? You cannot create a new Universe. The Furthest Ring is Vast and, before Feferi, there weren’t even Bubbles, so there’s the chance of wandering an Endless Void Forever. And Inconditional Immortality means you couldn’t even die, any way, ever. Just like the Black Hole at the core of the Dead Session, I like to think Dead Sessions are prisons- You’re fated for death, and if you miraculously survive, you will face an eternity of absolutely Nothing at all.

Of course, given the Kids went in to Caliborn with the Retcon Powers, and then sucked his Soul out into the Furthest Ring, he was spared such an eternal fate. Truly a Lord of Time.