Do higher Arena chests contain any lower Arena Rares and Epics?

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.