What's the right way to take a dog in a small plane?
My wife and I have a small dog that frequently travels with us.  We place her in a soft-sided kennel that straps onto the seat belt.  While this doesn't do anything to help in a sudden impact (i.e. crash), it does help to keep her from being bumped around in case of turbulence.  For larger dogs, I know that there are seat belt tethers out there.  My suggestion would be to strap them to a rear seat.  In either case, the big issue is to ensure they don't get in the way while operating the aircraft.

As for the hearing part, a company called Safe and Sound Pets makes a product called Mutt Muffs.  Their website is here.  My wife and I bought a pair, but our dog wouldn't have anything to do with them and they were returned.  Of course, your dog may be better behaved :D