If axe is attached to Arianne (Core), and her Action puts character into play, is there a reaction opportunity to put returned axe into play on a new character, or must I trigger axe's reaction immediately after character enters play?

No, because the character you put into play comes first before Arianne gets returned to hand.

Don't the pre-then and post-then parts of an effect share the same reaction window? 

Surely you could trigger Corpse Lake after Gregor triggered his ability to murder someone. 

Don't the pre-then and post-then parts of an effect share the same reaction window? 

Surely you could trigger Corpse Lake after Gregor triggered his ability to murder someone. 

hhhmm that's an interesting point to investigate.

I think the most comparable interaction this has is when Ramsay is brought into play with Arianne. The question that one is whether or not people can bring in Ramsay with Arianne and then trigger Ramsay's ability and sacrifice Arianne before Arianne's post-then effect takes place and returns her to hand. The answer to that interaction is no, by the time Ramsay can react to being brough into play the entire effect of Arianne's ability has already fully resolved and returned to hand. So I believe for the same reason, after both of Arianne's ability of pre and post then effect happens, Hotah's Axe is already in hand by the time Arianne's ability fully resolves and should be able to react to the character being put into play.

I guess the rules to nested ability sequences only applies to interrupts and not reactions.

As for Gregor Clegane, my understanding is that the discarded card still went to the discard pile and then moved to the dead pile as a reaction.

Is this a timing rule thing? What comes first? Arianne's 'then' or The axe's 'After'

Does it depend on First player?

If Arianne's 'then ' comes first, the axe would be in hand and therefore axe could trigger.

If Axe's 'After' would have to come first (the moment character enters) then you can't trigger it as it isn't in hand. I think...

As chriswhite pointed out, pre and post-then effects share a reaction window. Per FAQ section 1.8:

If the effect text of an ability includes the word "then," the pre-then and post-then aspects of the effect share a common reaction window, which opens after the entire effect has resolved.

Thus I believe the timing sequence is:

1. Player chooses to trigger Arianne's ability while she has Hotah's Axe attached

2. Interrupts to the character entering play

3. The character enters play

4. Interrupts to Arianne returning to hand

5. Arianne and the Axe return to hand simultaneously

6. Reactions to the character entering play, Arianne returning to hand, the Axe returning to hand

Since the Axe is already returned to hand in #5, you should be able to trigger its ability and put it into play on the new character during the combined reaction window in #6.

You can put the Axe into play. (My question was rhetorical.)   Movac is incorrect. Reactions to an element of an effect can be (and usually are) initiated and resolved after the resolution of the entire effect.  In this case, the reaction of the Axe is triggered after Arianne's effect resolves completely. 

(Edit: ystros interrupted my reply so his reply resolved first!)

@ystros

Thanks for laying that out . Clear as day now.