[Kallatari]

I seem to have found a partial answer, from which you can infer the remaining answers. Sharing it for everyone's info.

The rules state that a "dreaming" celestial can go to Heaven, and then return to his "sleeping" vessel simply by waking up (GURPS In Nomine, p.179). This clearly means that the vessel does not disappear when the dreaming celestial goes celestial via the Marshes.

Can he go back down a Tether in a second vessel? That's not answered directly.

However, there is a rule regarding the possession ability of Kyriotate (conveniently enough on the same page) that is relevant. A celestial whose vessel is possessed is forcefully ejected to the Marshes. If they make it back to Heaven (or Hell) and if they have a spare vessel, they can go back down to Earth in another vessel while the Kyriotate is still in possession of the original vessel. As soon as that original vessel is no longer possessed, if the owning celestial is in the Ethereal Marshes, he returns to this original vessel. If the celestial owner is instead in the celestial realm or in another vessel, the formerly possessed vessel dissipates.

That suggests that you cannot have two vessels in place at once (unless one is occupied by someone else), and any unoccupied vessel would disappear. So, I think it's reasonable to assume that a "sleeping" vessel would disappear if the projecting celestial owner returned to the Earth. But until that point, it definitely remains behind.

And it does seem to provide an "escape clause", allowing you to quickly retreat back to your vessel by waking up (which requires a Will roll).