Using Reserve Points While Unconscious

A character who is unconscious can only use reserve points if they are stable.

How Fast Are Reserve Points Recovered?

The references to you recovering your CON score + level, rather than CON modifier + level, are correct. Reserve points come back quickly. When multiplying this value (such as when a full day doing nothing but resting), add CON + level before multiplying.

Does a badly wounded character heal both hit points and reserve points at once when they rest?

Yes, provided both are still above zero. Characters on zero or negative hit points do not regain Reserve Points.