Ubuntu 20.04.2 LTS (Focal Fossa), with MATE DE. GNOME folders keep re-appearing in MATE DE environment, even after uninstalling GNOME

Solution 1:

To slightly rephrase and answer your questions:

Where do operating system directories with gnome in their name come from?
Answer this question by asking dpkg..for example:

dpkg -S /etc/gnome

This may give a list of packages. For a more specific answer, run this for specific files in the directories.

Can these cause your computer to heat up?
Only if they are running! Check with ps to see if they are. Sorting by cpu in top or system monitor may also be informative. Considering you only have 4G, sorting by memory use may also help. If you find running gnome components you are not using, you can just kill them, or you can dig deeper and prevent them from auto starting, see How do I manage applications on startup in GNOME 3? but also check the MATE autostart methods, as it shares some with gnome. Switching between MATE and Gnome may leave services from both auto starting when you log in.

Can you mix gnome applications with MATE?
Mostly -- the package manager will install gnome dependencies those tools need, but there may be gnome services that are also needed, so if you disable those in the above step, some of the applications may not have full functionality.

Looking at your screen shots, I am reminded that your web browser may actually use more memory and cpu than gnome, so addressing this may help. Hibernate or close any browser tabs you are not using. Check vivaldi's internal task manager to see if there's an especially greedy tab or extension. See https://vivaldi.com/blog/improve-performance-in-vivaldi/ for deeper coverage of this.