Hi HWTech. Your advice was really good! I went into Safe Mode. I clicked on one app which prompted a Rosetta installation to which I agreed. This was a successful operation! Next it asked to update my app software. Thereafter, the non-Apple app was able to open! Then I went on doing the same with every non-Apple app, and they all worked! The trick was to get Rosetta installed first and it's only done once.
The other interesting point was that I could not even update my macOS before I got into the Safe Mode, as I got the same message about not being connected to the internet. Now I can get to update my macOS even outside of Safe Mode.
Thank you for really giving me an excellent advice! Much appreciate it!