Throne of glass fae Immortality - How does it work?

In throne of glass, the Fae are said to be immortal(can't age, but can still die), but then is Celaena immortal? Even if she stayed forever in her human/animal body? Or do they age when they turn to their animal form, and stop when they become fey? Do their gained age reverse when in fey form? If they don't age in their fey form, how did the 2 Fae sisters die? I know they fell in love with mortals, does that remove their immortality? What about their magic...

I might have missed something in the books, so if anyone who know can point it out? Thanks!