Bitwarden is more of a direct replacement to Lastpass, but better and open source with self hosting options (see also Vaultwarden). Keepass is a safe stored and managed by you, no syncing or online services are included.

There are extensions for it.

Personally, I use syncthing to send my keepass database to the various devices I need it on.

From my personal experience switching from last pass, yes, it has been a very good replacement. Inside of LastPass, there is actually a way to get a spreadsheet with all your account details like username, password, URL, notes, etc and then import that into keepass. Once you have done that, you obviously delete the spreadsheet since it's in plain text. However, I will say that keepass Is not right for everybody. You have to control your own database file and if you lose it, then you are shit out of luck. So you have to make sure that it is properly backed up and synced between devices. I use a quarterly backup strategy and a flash drive to accomplish this, but your choice may vary. However, using a cloud service of any kind to put the file on is probably not a great idea. If you must use a "cloud" I would highly suggest syncthing since you remain in control at all times.