How remove Sakura R-727II range hood to clean inside

Never seen one like this - BUT have installed many a hood in cabinets where the vent did not end up exiting the center of the cabinet. a combination of adjustable elbows will offset the vent if you have at least 15" height in the cabinet. As a last resort flexible aluminum duct could be used but only if you have no other options.

How do you know that the wire is on the same circuit as the lights??

I can clearly see the wall so you can open it.

If you are not opening the wall how are you installing a dedicated circuit for the new microwave and where are you putting the outlet for that?

"Can in install a box where the wires come through"

YES. You can install a "old work" box there, (made expressly for retrofit). Cut a hole around the wire carefully just big enough for the box, use the box as a template and trace around it with a pencil. Run the wires in to the back of the box and cap them with wire nuts,insert the old work box and tighten the screws that pull the tabs which old the box to the drywall, put a solid cover on the box.