Abilities » Stance Change
If I use Aegislash's "Stance Change" Ability, to switch for another Aegislash, can I then use the new Aegislash's "Stance Change" if I want to?
No, even though you have switched to a new Aegislash card you cannot use Stance Change again because the effect of using it remains on that Pokemon.
Source: XY FAQ (2014-02-06), TPCi Rules Team (2014-02-06)
Abilities » Stance Change
When using Aegislash's "Stance Change" Ability, do I have to switch the Doublade & Honedge underneath it too?
No, you only switch one Aegislash card for another Aegislash card.
Source: XY FAQ (2014-02-06), TPCi Rules Team (2014-02-06)
Abilities » Stance Change
When using Aegislash's "Stance Change" Ability, what do I do with the card after I switch it with an Aegislash from my hand?
The Aegislash that was in play goes into your hand.
Source: XY FAQ (2014-02-06), TPCi Rules Team (2014-02-06)