Once demoted to a lower arena, then promoted back, will the arena offer return

As far as I can tell, no.
The only reason to open the Arena X chests would be that you're out of Arena X+1 chests. Maybe if the higher chest is a gold one and you're saving it for the night or work day.

You many want to clear them out of the way at some point to make room for better ones, but if you're constantly getting just as many chests as you need, I'd postpone opening the lower level ones.

Of coures, they are in your inventory. I'll propably open them at some point anyways. Maybe that time you fall to the lower Arena. Maybe that one day you just don't feel like playing.

You get officially demoted when you reach 50 trophies below the threshold for a given Arena. You will receive an in-game notification when this occurs. Similarly, you get promoted when you reach you cross the threshold listed for a given Arena. You can view the thresholds for all Arenas at any time by tapping the picture of the Arena you are currently in above the "Battle" button on the Battle tab.

As others have noticed, you can visit the Arena above or below your Arena, if you are close to the threshold and get matched with someone who has qualified for that Arena. Chests you are awarded for victory will be locked to the Arena where the match took place.

Note however, that the Free Chest, Crown Chest, and Shop will all reflect Cards and Gold according to the Arena you are actually qualified for. This is the main difference of being on one side or the other of the threshold.