For 10 mile rides, I don't think you'll notice a difference between 16" and 20" wheels. Especially if the roads around you are in good shape. Maybe after 30 miles you'd notice.
Dahon and Downtube are probably your best bet in your budget range. If you don't ride up hills, try to find a new Dahon Boardwalk S1, which has one gear and is around $200.
I think there's a very good chance that if you bag up your bike, you can bring it into a library. However, the best thing to do is talk to the librarians and ask them about it. I don't think you want to get on their bad side, right?
Also, sooner or later you'll probably have to lock up your bike outside. This will obviously be less frequent (and therefore safer) with a folding bike; plus the bike will take up less room at home.
Regardless of the type of bike you get, I'd get a very solid lock and make sure your credit card will cover you in case of theft.