Is it true that eagles are able to literally 'soar above a storm' and are the only birds capable of doing this? If so what is it about their design that enables them to fly above a storm?

Eagles can fly at a great height so it may be possible for them to fly above low clouds. However, most birds are more likley to take shelter when there's a storm.

There are also many species of birds which can fly higher than eagles, including several species of waterbird - the absolute altitude record is held by a Ruppell's griffon vulture which was hit by a plane at 37,900 feet, so if eagles can fly over a storm, so could various other species.  However, the main reason predatory birds like eagles fly high is to spot prey - if there's a cloud in the way, it's not much use them being up there!