Do higher Arena chests contain any lower Arena Rares and Epics? I'm on Arena 3 now but only have 1 of all 6 unlockable Epics at this point. Should I stay at 3 and keep grinding up, or should I fall to 2 and 1 to collect any epics I'm not getting currently?
Yes they do.
I am on Arena 4 right now and I received an Epic Baby Dragon (which is available since the very beginning) in my last chest. I also often receive Rares from lower Arena ranks from any chests I open.
The opposite would be silly too, people would have to lose on purpose to stay in a certain threshold in order to receive some cards, it wouldn't make much sense.
As I mentioned in my answer on a similar question, it is indeed possible to get cards from previous arenas in higher tier arena chests.
In fact, it is generally better to advance as far as you can in order to obtain cards from previous arenas. Even factoring in the lower drop chance and the longer cycle time of the shop, higher tier arenas will get your lower tier cards leveled up faster.
I put together a card upgrade calculator that does all the math for you - you can play with the various settings and see if you reach the same conclusion.
Notice that the trophy threshold is (number)+ for all arenas, meaning you can unlock the cards from those arenas once you've unlocked them