How can I see only the version numbers of each item in my library? Because now I have many games and it is tedious to see them one-by-one to check the version. Is there a way to search through faster? I need this feature because they get updated and I only see new updates in a weird way, only if it is installed and I keep the desktop clean of the launching icons.

There is no such a feature implemented I'm afraid.
Yepoleb's GOG Database seems to be the fastest way to check for latest updates.