You have not provided near enough information to actually answer any of your questions fully. With the information provided:
There are two potential issues here. I’ve seen some reports that System started rules do not work in Rules DSL .rules files, but I thought that was fixed.
The second issue is what rossko57 points to. System started now properly triggers when the system starts, not merely when a .rules file is loaded.
Need more information. MapDB works and works well for me. I don’t think you can bring over the MapDB from your 2.5 instance to the 3.0 instance so you’ll need to repopulate those Items.
Anything more will require seeing how you’ve set up the .persist file and what you’ve done to try to test it.
OH 3 implements it’s own authentication and authorization. You need to modify the config. See OH3 with NGINX Reverse Proxy and Authentication and look at the full thread because there are different options.