MacOS – Keyboard shortcut to move Window to next Desktop

"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!

There are many, many different sharewares out there that let you move windows around using shortcuts. Just look for window manager in the Mac App Store. I would recommend BetterTouchTool. It's a freeware that let's you customize just about anything. For example here is the shortcuts I have created for myself :

http://blog.boastr.net/