Once demoted to a lower arena, can you still get the higher arena cards

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.

In the case of Free Chests and Crown Chests, the content of the chest will depend on the moment you open it. So it will use your current arena level. This seems to be confirmed in this forum thread :

Next, whatever arena you are at when you receive the chest, that is the level it will stay regardless of what level you are at when it is opened. Crown chests and free chests are based on what arena you are at when you open them.

It also seems to be confirmed in this guide :

The kinds of cards you can get depend on the arena you are in. For example, if you are in arena 3, the chests can contain cards available in arena 3 and below.