They probably would look shadowy, possibly something like what they looked like in the hobbit movies when they fought the white council
I think elves have a choice whether they want to come back or not
I would say because that was the end, if Frodo and Sam failed that would have been the defeat of the Men of the West and all free people of middle earth including the Eagles probably, so they were invested in how that last battle played out and probably decided it was best to help, and when Frodo and Sam were successful they got trapped and only the Eagles could get to them to save them so I'm sure Gandalf was able to convince them it was the right thing to do to go get them
I first heard of LOTR when The Fellowship of the Ring came out in theaters, I was in 8th grade and went and saw it and loved it, after that I checked out The Hobbit from the library and read it and loved that, then read the LOTR books, I saw the next two movies in the theater my freshman and sophomore years of high school
The Eagles are sentient beings, just like men, they're not a beast of burden, and they are a very proud race, just like men can choose not to operate a taxi service, they choose not to fly anyone around, even though they could