Guys, thank you so much for your kind words!

Today, my new nut arrived in the mail. I found a TusQ nut that works, I just had to sand it thinner and remove some width on the left and right.
I noticed that the nut slot on the guitar wasn't perfectly level, so I filled the bass side of the slot with wood filler, and sanded it level when it was dry.

Now the new nut is in place and it's already a big improvement. I was able to get all strings to a decent height. Still a tiny bit of fret buzz but nothing compared to how it was before. I think I'll leave it this way, it works well. Maybe I'll change the strings on the bass side to a thicker gauge but let's see.

Before taking any drastic steps and alter the neck, I'm trying to get used to it first :-)
I've been playing it almost daily and it's already better now, I only have problems reaching the low E and A with my pinky, which is too short. I have to twist my wirst a bit more than I'd like to, but I've been spiderwalking the neck and I hope it helps me to get used to it faster.

And while I still admire the feeling of the tru oil, I think it's too delicate as a finish. I know it can be fixed really easily compared to other finishes, but I can already see many fine scratches just from gently using the guitar. I know it will take a bit more time to completely cure and harden, but if there isn't a miracle happening, I don't think the tru oil is very durable for a guitar that's getting played. I'm not rough with my stuff (I think), I never dinged it, always put it on padded surfaces etc, but still, there are some marks.

I'm going to hang this guitar on the wall for a few weeks without playing it much, so I can see if it hardens a little more.