Don't know if it was wise from my side to do that without making a backup
but I searched the registry and deleted every entry that contained word "demigod". One of them contained "Patch level" 5 (I uninstalled the game before making wise decisions).
I'm reinstalling now, will update this post after reinstalling and re-updating.
Update: I un-installed the game, deleted all registry entries that contained demigod (please make backup first before doing something like deleting registry entries) and re-installed the game after it (re-downloaded and re-installed). I can't test pantheon games at the moment but other character seletion screens work absolutely correctly. Thanks to Zekail for the idea and thanks to developers for their fixes.
Update2: most of issues I ever had seem to be fixed now, but UncleanBeast, which I use, still keeps roaring when I start a skirmish game (because I never leave character selection screen). Its fine with me once, twice, maybe, 3 times. But since it takes me at least 3 minutes to connect, I don't really enjoy the continuing roaring. Would be nice if it stopped as soon as I hit "Fight" button. Lag problems remain but now multiplayer game feels almost like SP with small and rare lags, thanks for that! I think it accumulates lags - the longer you play the more it lags... but I could be wrong.
In either case I hope this information helps other people: you need to clean up the registry if after re-installing the game you still have issues, my registry had info on the version of the patch installed after uninstalling the game (I have a localized version and I uninstalled it and installed original version, I think the bug came from that, after un-installing original and cleaning registry and re-installing original it seems to finally work good).