Auto-center mouse cursor on specific display, based on used trackpad I have a MacBook connected to an external display + external keyboard and magic trackpad.

Because the laptop is placed on the table, while multitasking between the two displays (internal and external) I often find it useful to use the built-in keyboard and trackpad from the MacBook when I want to focus on laptop's screen.

When I start typing with the laptop keyboard or using the built-in trackpad, is there way to:
1) Center mouse cursor on laptop display?
2) Make app switcher (CMD+TAB) icons show on laptop display without having to reveal dock on that display first?
3) Do all or some of the above on external display when I switch back to external input device

This might sound crazy, but still curious if it's possible :-)
There are ways to center the cursor in either built-in or external display.

Apple SE: CatchMouse.

So first press a hotkey to switch to the desired display. The cursor is now on that display and app switcher is shown there as well.