Are you allowed to swap Ability Cards between Obelisks at any point?
Nope, the ability cards are permanent.
Are you allowed to leave a unpowered (thus unused) Ability Card at the base? For example: when bringing a Obelisk into battle with two Ability Cards attached to it, one powered and one not? I guess not, as each Ability Card has to be attached to an Obelisk Card?
You have to take along any ability cards that are attached to the obelisk to use that obelisk in battle (they are permanently attached to the obelisk) However, you can (and MUST) power ability cards individually if you want to take advantage of said abilities, even if they are on the same obelisk.
I hope this answered your question.