noglider 

Yes, some have switched to coaster brakes instead of rim brakes. It's still a good idea to keep a front caliper brake. You may find that the coaster brake is more than adequate for stopping, but if your chain falls off, you'll need another brake. This happens!

For brake levers, use the search term "aero."

You can't run the shift cable through that last bit of housing because the hub doesn't have a cable stop built in, but you might be able to use that last cable stop on the frame. It's facing the wrong way, but there's a chance you can make it work.