I am going on a school trip to Disney next weekend, and I just realized that since we are leaving Friday around noon, driving through the night, and we don't get the chance to change, shower, or anything until Saturday night, which means I will have to bind constantly from 7am Friday to around 10pm Saturday.
I could switched into a stretched out binder at night and change back in the morning when we get to the park. but then I would be carrying around a binder in a bag while spending the entire day at the park.  And I want to limit the amount of stuff I have to carry around.

Also, when it comes to the bathrooms, is there a way to know where family bathrooms are located? I've looked for maps and didn't find any, and last time I went to the magic kingdom I never came upon a family bathroom. 

To be honest the binder shouldn't really be an issue unless you have a lot of problems with them. I've had friends over for a couple of nights in a row several times, and been forced to wear my binder for a few days in a row, and it's never hurt me badly. Sometimes my chest has hurt a tiny bit after (it's not good to wear them while sleeping, but doing it once or twice won't kill you) but never anything worse than that.

Not sure how up to date this is, but it looks like they're called "companion bathrooms".

http://en.allexperts.com/q/Walt-Disney-World-234/Single-dad-travelling-4.htm

I'm sure they have family bathrooms at Disney.  Binders can fold up pretty small.  At least the one I use (underworks medium style 997) does and I can stuff it into a fairly small bag, say three by three inches.

I bind over night all the time when I spend the night over at someone's house.  I've never had a problem.
I say you know your own body.  If you're in pain or having trouble breathing, you got reason to worry.   If you feel fine, I wouldn't worry.

Like the others, I've never had any serious issues other than some discomfort sleeping in a binder. I don't know about the family rooms, but you could call and ask them, I'm sure.  Are the handicapped toilets built into the individual men's/women's rooms or do they have a separate toilet like they do here in South Australia?