I will focus on your first question, but the reference I will include also will help in resolving at least parts of your other questions.
(1) By a theorem of Marek, if there is a second-order definable well-ordering of the continuum, then as soon as two countable structures are elementarily equivalent in second order logic, then they are isomorphic.
(2) On the other hand, there is also an old theorem of Harrington that says that every model of ZFC has a generic extension to a model of ZFC in which the continuum has a second order definable well-ordering (more precisely, a $\Delta^1_3$ one).
(3) Finally, it is known that there is a model $M$ of ZFC that contains two distinct countable ordinals (countable in the sense of $M$) with identical second order theories (in the vocabulary of linear orders).
So, by starting with $M$ as in (3), and then invoking (2), by (1) we arrive at a generic extension $M[G]$ of $M$ in which two countable ordinals that used to have the same second order theory now have distinct second order theories.
You can find details about all of the above in this manuscript by Lauri Keskinen (which I believe is his Ph.D. dissertation at ILLC Amsterdam).
You may also find my answer to this question. to be of interest.