Your foot shouldn't hit the rail. It's good to be close to the rail, but...your foot still shouldn't hit it. That's a bit too close. I would tell her in private, screw what your Dad says. :) You're not being a baby, you have to speak up! Getting pinched by the rail isn't comfortable at all. If he gets angry, so be it...or like 'ouch! I got pinched!'. She shouldn't make you ride THAT close. Especially if you're in paddock boots that shouldn't happen. You could try a different horse, but it sounds like it's the instructor not the horse that's the issue.
As for letting them ride sick, I guess it depends how sick...I've ridden with a cold before, but once I wasn't contagious/well enough to ride. If I'm practically dying with a sore throat/runny nose, nope. People shouldn't go to the barn if they're sick like that, but to each their own. If you aren't happy with the facility/instructors maybe try to find another place? I know it's probably hard to find another place though. :(
Ride more, worry less.