Welcome to the forum!
Ok, first I need to be clear about the problem. Do you mean that only the new extensions which you installed are not opening? Can you open other extensions, besides what you installed?
I appears you installed to the correct location. It's curious that they didn't show up unless you installed to both places. Either place should be enough. So I'm not sure what's happening there.
If the files you downloaded were zip files, did you unzip or extract them? The PY and INX files need to be placed directly into the extensions folder. Inkscape can't find them inside zip files.
Those entries you posted from the errors log -- did you only post those which pertain to the extensions you're using for this project? Usually there are a lot more. Plus, it appears that inkex.py is the culprit in every case. Could you have somehow forgotten that file, or left in another folder or on the desktop or something?
I'm not clear where you sent a message regarding python. You should not need to install it. Inkscape provides what it needs to function. Although it seems like I heard about an issue with not having the right version of python. I'd have to do some research, to remember what it was about.
We have heard from someone else that the J Tech Phonics extension doesn't work, even if you could open it. There's another message somewhere in this forum about that.
Ok, I've just tested some of the gcodetools extensions, and they are working for me. I'm highly suspicious of the inkex.py file being missing. At least until we get more info from you.