Sometimes my character (mostly Gork) disappears in multiplayer games after entering a next level (around 5-7).

I believe this is due to lag. When you have lag, players do not accurately see an up to date location of their partners' heroes. What's probably happening is that your friend is pressing the elevator button because he thinks you're on the elevator... but you're not there yet on your own computer. If enemies slow you down, your location is even further desynchronized. The best workaround I can recommend is having all players say "READY!" before the crystal runner presses exit.

No... my charakter was always already in the elevator (on my PC and his) before my friend pressed the button.

My charakter even showes up in the transition scene between the levels on my and his PC.

But when the new level was loaded the charakter was gone.

I'd completely forgotten this, but once I did have an issue where I was playing with friends and on a new level, one of us was invisible. The character was still in the top right, could still be selected by the player, and they could even run around, open doors and attack things, but they were completely invisible for everyone. Can't remember who/when cause it was a while ago now, and only happened once.