Some more info what I tried at my free weekend to solve the problem:
updated MacOS to Monterey 12.4
restarted the Mac multiple times
downloaded DBeaver (one of the apps that didn't run, see initial post) via browser: https://dbeaver.io/files/dbeaver-ce-latest-macos-x86_64.dmg
started by double-clicking the downloaded .dmg file in the browser and click the app icon (with and also without moving to /Applications): dock icon starts bouncing, Warning "CoreServicesUIAgent Connection from process XXXX does not have the required entitlement com.apple.private.iscsuia", dock icon stopped bouncing, app does not start (same behavior as in my initial post)
opened the .dmg via terminal: open dbeaver-ce-latest-macos-x86_64.dmg -> same behavior
started the .app via terminal: /Applications/DBeaver.app/Contents/MacOS/dbeaver - nothing happens, no output, terminal hangs forever.
checked the executable (x) flags via ls -al - it is set
checked and removed quarantine flag in DBeaver.app: within /Applications/DBeaver.app/Contents/MacOS: ls -al@, xattr -d com.apple.quarantine dbeaver, as well as xattr -r -d com.apple.quarantine DBeaver.app/ - done, but app doesn't start either
downloaded the app via curl: curl -O -L https://dbeaver.io/files/dbeaver-ce-latest-macos-x86_64.dmg and started via Terminal "open dbeaver-ce-latest-macos-x86_64.dmg" or Doubleclick in Finder -> works as expected.
downloaded again via web browser, removed the quarantine flag via "xattr -d com.apple.quarantine "dbeaver-ce-22.0.4-macos-x86_64 (1).dmg", clicked .dmg in web browser and clicked the .app without moving it into /Applications -> works! If clicked this .dmg in web browser, moved the .app to /Applications and run it from there, it does not start.
mounted this .dmg via terminal hdiutil attach "dbeaver-ce-22.0.4-macos-x86_64 (1).dmg" and started via terminal "/Volumes/DBeaver Community/DBeaver.app/Contents/MacOS/dbeaver", the app started once normally and printed a lot of infos. ok I said, then I wanted to do the same with the .dmg with the other .dmg with the quarantine flag bit being set. But since then I'm unable to start any app successfully anymore: neither per browser, nor by terminal (open, curl, execute)! What's going on here? Apple seems to store hidden information which makes it impossible to debug and reproduce.
btw., other apps seem to have the same problems on Bug Sur, e. g. here: https://github.com/beakerbrowser/beaker/issues/1861
I didn't try but I'm sure that all other apps shows same behavior.
In the result, the quarantine bit seems to be one key of the problem. But, as written above, also downloads without that bit set being set didn't start anymore.
Maybe someone could try to download one of the .dmg files in my initial post and try to start the app in Monterey and post here if it works? Thank you. I'm searching now for weeks for a solution and still don't know how to build a new version for our customers who are waiting for a solution.