By Zariski's main theorem, an injective morphism of smooth complex varieties is an open embedding. From this it is easy to deduce that an injective endomorfism of a smooth complex variety is an isomorphism. Hence RP and JC are equivalent.

Oops, I am sorry and you are right, $E$ is not defined over the base field. So this is not a counterexample!

Try "Varieties with isomorphic or birational hyperplane sections", by James McKernan.

I suspect that the author of the post intends to assume that the generic automorphism group of a stacky curve is trivial. In this case the category is in fact a 1-category.

You could choose a different embedding for a projective space, and then it you would not be true any longer.

For Deligne-Mumford stacks, this is claimed as Proposition 4.4 in the original paper of Deligne-Mumford "On the irreducibility of the moduli space of curves", but without proof. I have never been able to figure out why this should be true.