This could very well be Voyage of the Basset. It was written by James C. Christensen, with Renwick St. James and Alan Dean Foster. Later, the story was adapted into a novel, and the novel continued into a series, by Tanith Lee. It is that, I think, that you remember reading.
- Voyage of the Basset sounds similar to Voyage of the Dawn Treader. Indeed, there are some common themes between the two (a magical chain of islands).
- The first book in the series features Pandora's Box.
The second book features the Unseelie Court.
Gwen Thornworth has dreamed of dancing with fairies since she was a
little girl. When the Bassett arrives to pick her up, it looks like
her dream will come true! Gwen's twin brother, Devin, reluctantly
comes along to keep an eye on her. And he has to do just that when
Gwen is lured away to the dark halls of the Unseelie fairies. Can
Devin rescue his sister or will he also be caught under the spell of
the Raven Queen? This second book in this thrilling new series that
combines magic and mythology will captivate readers.
I don't know whether there is a thunderbird, but there is a Fire Bird.
Travel is via rainbow:
"Our direction really is to be upward. We must ascend the rainbow."
Another possible confusion is The Chronicles of the Imaginarium Geographica.
It is clear that James Owen was inspired by Voyage of the Basset, since Here There Be Dragons, the first book in the Chronicles of the Imaginarium Geographica, features a magical ship which can cross to a magical Archipelago The ship is crewed by fauns, there is a faerie isle of some sort near the entrance to the Archipelago, and Pandora's box has a prominent and sinister presence in the plot.
Yet to my knowledge, James Owen has never acknowledged that Lee or Christensen were among his inspirations.