Let's say my opponent has an active dragon pit out. I have the Lannister / Stark Hound in my hand (the one that jumps if a character's strength gets reduced). Will playing him out immediately cause him to jump back to my hand?
Yes. The reason why is that his passive "after STR is lowered" effect is initiated by the new application
of a -X STR modifier to a character. The continuous effect from Dragonpit applies such a modifier as soon as (or "as part of") that character coming into play, so there is a "trigger" for the passive.
Stannis's Northern Cavalry is out and there's a king's pavelion in play. Are they immune to being discarded / killed while standing? What if they kneel during the phase?
Like above, a character's STR is only considered "lowered" if a new -X STR modifier is applied to it. Similarly, a character's STR is only considered "raised" if a new +X STR modifier is applied. So, if you start the phase with SNC standing, the bonus from the Pavillion is not "new" because it was (continuously) applying before the phase even began. If SNC kneels during that phase, it still hasn't had a new -X or +X STR modifier applied
to it. It just had an old one "wear off." (Losing a +X STR modifier is not "lowering" the character's STR, even though the total effective STR goes down). But
, if they stand again, the reapplication of the +X STR modifier from the Pavillion will be considered a "new application" because the modifier didn't apply, and now it does.
Finally, King's pavelion is out, so is my hound. Will Hound bounce if he kneels?
See above. Losing
an old +X modifier is not considered "lowering" the STR. Only gaining
a new -X modifier is.
The reason this works the way it does is because STR is, at all times, the total of the printed/base STR and all applicable modifiers. You start from scratch each time you calculate STR. Since you do start from scratch each time, the current STR check has nothing to do with the previous STR check, so the difference between the two cannot define "raised" and "lowered" STR. That means that only the introduction of a new modifier into the check can count as raising or lowering.