As I rank up in ranked games, do my casual opponents also get tougher? That is, does my rank have any bearing on the casual matching system?
Casual and ranked play use separate ratings for matching you with opponents.
The blog post that came with the latest patch also says this:
Casual Play will not reset on a monthly basis like Ranked Play does—Casual Play will continue to match you with other players of similar skill.
So, casual matches should still match you up with players of similar skills (however Blizzard chooses to define that) regardless of your spot in Ranked Play. This seems to make sense, since ranks are currently set up to reset at the beginning of every month.