Hi crinrict , in the meantime, I continued with my save, so for the testing and taking my mods out, I had to make new save. Weird thing is, everything works now in the new save AND the one that had issue even with the mods back in game. So it might have been the specific sim issue? (I cannot test it specifically with her now).
But just to give more info about the problem I had yesterday. It seemed to be only moving into dorms issue, because since I had one sim only, the "split household" did not appear when moving through phone.
The sim technically got enrolled in the uni - today while testing in the save where the sim disappeared after I closed the game when the issue occurred, I found her again - she was just standing in front of her old house. Weird thing is, she was not anywhere to be found in the manage households, but she was in the world. I added her to household I was testing with and she was in fact enrolled to uni.
And it got stuck after I clicked the check button. I could move money, I could click on the sim, but not move her to the other "old household" side, and I could click and unclick the "sell everything from the house" (or some similar type of message, I'm sorry I'm translating it from my language).