After chatting with tech support, it seems that DRM is fine on its own but there are always possibilities of connectivity issues or some registry errors. I have to clean up everything (possibly using Revo) and will try to start from ground zero or as close as reasonably possible incluing graphic drivers.
Nothing works as the problem seems to be on the server side. The old temporary fix works though:
1. Set Origin offline
2. Start ME3, click on Connect to EA servers, cannot connect
3. Go online in Origin
4. Rapidly click on Connect to EA servers
5. Rapidly go to multiplayer. Play as much as you want either hosting or off host.
The only problem is that it is impossible to accept invites from friends as you are teleported to main menu and the game crashes right after that. Otherwise, everything seems to function normally.
It is probably a similar thing to Retaliation glitch/bug. But unlike other people noticed, deleting/moving Retaliation DLC folder does not help in my case.
