Note that #7 is not actually true anymore, and actually hasn't been for quite a while.
A paper from Mehdi Tibouchi [1] shows how to represent points on essentially any curve as uniform random bitstrings.
Another paper by Aranha, Fouque, Qian, Tibouchi, and Zapalowicz [2] gives an even more efficient construction for curves over binary fields.
(As an aside, there are really good reasons not to use curves over binary fields. Discrete log has recently been getting easier much faster over GF(2^k) than over GF(p).)