If I was in arena 4 but got demoted back to arena 3, will I still get the arena 4 cards?
Nope, your chests are tied to your current - not highest - arena level. If you make it to the top and then fall back to arena 1, you'll get arena 1 chests with arena 1 cards.
While what agent86 said is true, if you have any chests over from the higher arena, they will still contain those arena's cards. So yes, you will still get arena 4 cards but you will not get arena 4 chests. Therefore, the only way to get the arena 4 cards would be if there were arena 4 cards in the shop before you were demoted, or if you had any arena 4 chests.
In fact, the answer is yes (to your title, at least), but only in a very specific circumstance. As I mentioned here - the Arena you play in determines what the awards will be for either player if they are the victor. You can therefore get a Chest from a higher Arena if you are high enough and get lucky (both by getting matched to a higher Arena, and winning that match).
No. But I have discovered that if you unlock a card from a arena, and then fall back down, you can still get more of that card.
No. what ever arena you are in, is what arena cards your gonna get... If your in arena 4 and drop down to 3, you'll still get your chests from 4; and then once your out... Your gonna get lvl 3 arena cards only;)
What everyone answered is true. However, you can also still request the higher-arena cards if you have already recieved them.