How can I create keyboard shortcuts to move to previous / next full-screen app?
"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!