Q: Is Celaena immortal?
A: Not yet. As far as the series goes (up to Empire of Storms), Celaena has been growing like any ordinary person. But this was because she hadn't reached her time yet. As Maeve says to her in Chapter 71, "My sister Mab's line ran true. The full powers, shifting abilities, and the immortality of the Fae. You're likely about five years away from Settling."
As we see from the Throne of Glass Wikia here, "All pureblooded Fae, and some Demi-Fae, go through the Settling - with their bodies attempting to be locked into immortal youth. It is a rough process, their bodies and magic needing months to adjust to the sudden freezing and reordering of their aging process."
Q: Even if she stayed forever in her human/animal body?
A: Yes. Eventhough her parents were only part-Fae and pure human, Celaena received the full Fae bloodline from Mab, so she is in essence just like any Fae we meet, except she just hasn't Settled yet.
Q: Do they age when they turn into their animal form, and stop when they become Fae?
A: No. As we see from Rowan's story, he 'went mad' for about 10 years, often staying in his animal hawk form after Lyria died. And he never aged. The only reason we see Celaena still aging, whether in her Fae or animal (human) form is because she hasn't Settled yet.
Q: If they don't age in their Fae form, how did the 2 Fae sisters die?
A: The answer is only speculated in the books, but it is evident that they weren't immortal when they died. The 2 Fae sisters Mab and Mora fell in love with mortal men and gave up their immortality to live and die with their loved ones. Some stories say that Maeve forced them to give up their immortality when she found out they had fallen in love with Mortals, other stories say that the 2 sisters had willingly given up their immortality so as to escape being with Maeve forever. The true answer is not confirmed yet. But it is clear that when Mab and Mora fell in love with mortals, they became mortal as well. So the 2 sisters died of normal old age.
However, a Fae, while immortal may also die after living for a very long while. Again from the Wikia "While the Fae may not grow old after they have Settled, after centuries of living, their power may weaken and the Fade. After Fading, Rowan's parents passed into something called the 'void', which may be a Fae concept of the afterlife."
Q: What about their magic?
A: Not mentioned (as far as I can remember) in the books. But seeing as we have humans and Demi-Fae (part Fae, part human) who have magic, and not just the Fae who can use magic, it makes sense that the possession of magic is not restricted to Fae. Thus, Mab and Mora would likely have retained their magical abilities and (maybe) their shapeshifting abilities. But they would just have continued aging like mortals.