When you upgrade from one Leap version to the next (in your case 15.5 > 15.6) the zypper dup (when done correct with --releasever 15.6, or after chaning the repo URLs manualy), will install all packages from the 15.6 repos “over” those that you have from the 15.5 repos. And you should end up with a correct working Leap 15.6.
I have no idea what you are referring to. When you ask about programs that are not from the openSUSE Leap standard repos, then you can only check with the source where you got them from, or you can test them.
That will depend on the answers you got (either by asking, or by seeing that the source has a 15.6 version of the package, or after testing).
Thus yes, all software you installed from elsewhere needs attention.
And it is here that the now available beta version of 15.6 can help. Use it to check your most important non openSUSE applications. Either on a left-over system, or in a virtualization environment.
If you have installed 3rd party apps which don’t provide an update repo, you are responsible yourself for checking that this apps stay up to date. If the app provides update repos which can be added via zypper/yum, they get updated with zypper dup. A good example is Google Chrome. It has an update repo which will update Chrome when you do a zypper dup…