Your EXE file associations have been damaged. This is a pain to fix but you seem skilled enough to clean it up using regedit. EXE association corruption is most frequently the result of a malware infestation and will prevent the execution of any program by the Windows shell. Corrupted EXE associations do not, however, prevent task manager's run command from working. You can use task manager to start regedit. Press ctrl-alt-del and start task manager. Select file > new task. Type regedit in the dialog box and press enter. Navigate to HKEY_CLASSES_ROOT\exefile\shell\open\command. Change both the "(default)" and "IsolatedCommand" keys to read
WITH the quotes. Now navigate to HKEY_CURRENT_USER\Software\Microsoft\Windows\CurrentVersion\Explorer\FileExts\.exe If you have a UserChoice subkey (check in the left pane), delete it. This should fix your system without the need to reboot. If it doesn't then you have other problems. Run Malwarebytes immediately to clean up whatever malware infestation damaged your EXE associations.
Apologies for the lack of paragraphs in my post -- for some reason the forum won't accept line breaks in that post as long as I have the HKEY_CLASSES_ROOT key in the post.