MacOS – Keyboard Shortcut for moving left/right a space doesn’t work

"In addition, I would like to change the shortcut in 'Mission Control' which 'Move left/right a space', but doesn't seem like I can change it."

I believe I can help answer this part.

At first I thought that a drop down would appear, but it does not. All you have to do, if you want to use keyboard shortcuts (not sure about trackpad or mouse) is hit the key once the item is selected.

Above is a screenshot of the menu with default settings.

Below is a screenshot of the menu with moving to the right changed to "."

So, to recap: Highlight item to change, double click on current shortcut symbol (to highlight it), immediately hit new key for the shortcut.

Hope this helps some!

I've noticed that certain applications seem to trap the gestures, even when they're no longer active.

The only time I've been able to 100% narrow down something trapping various shortcut keys was with either Apple Remote Desktop (the built in screen sharing functions), or Microsoft Remote Desktop.

Even though I clicked outside of the app and started using something else, something caused notably my ctrl+←/→ to stop working.

I unfortunately do not remember the specific steps I took to resolve the issue. Whether it was explicitly sending focus into the remote connection window, and then (with the mouse only) explicitly out of the window, and onto another app that is not just the Desktop, might have fixed the problem. Otherwise, I may have just closed the window in frustration.

Either way, this issue is one that you may have to do some personal diagnosing to figure out. Check for any running apps that are notably not native (Java, AIR-based, etc.), or have some other form of deep integration (trapping keyboard events and sending them to another computer is pretty deep), that may be causing this issue.