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Would it stay in high speed motion until interacted upon by something else (meteoroids/dust/gravitational fields)?
No, we've seen that warp field/engine failures will cause the ship to drop to sublight, if they don't slow to stop, rather than maintaining their current speed.
Or would it disintegrate under this kind of speed?
If it crosses the boundary of the warp field, it might be disintegrated by the resulting stresses, since warp field boundaries can be pretty rough.
I certainly don't think they'd continue to move at a high speed. Inertia wouldn't really apply, because the objects were never really "moving" at all. Space was moving around them, but they were stationary relative to the space inside the bubble.
So I'd say that if they survived crossing the threshold from the space inside the bubble to the space outside the bubble, they'd basically instantly become stationary.
The question is can you survive exiting the bubble like that, or would the bubble's edge tear you apart. But I think we might actually have an answer to this, from Discovery's last season. Didn't Burnham survive exiting that enemy ship's bubble in that first episode, before Discovery beamed her back aboard?
I just revisited the scene from "Red Directive" in which Burnham is riding on the top of Moll and L'ak's ship - at the end of the sequence, she leaps from the ship's hull as the warp field collapses. When she and the ship enter "normal" space, they both seem fairly stationary. Burnham is tumbling a bit in her EV suit, but she doesn't seem to have a lot of velocity.
The question is "stupid" as warp as described by Star Trek is somewhat unphysical.
Warp drives work by bending space and hence shortening the distance (in "meters") between you and your destination. Hence you can fly at a leisurely pace to get there. It is very much unclear how the space around objects changes. If they do not evade they would change "shape" as they approach your warp bubble. When they are inside I would expect you to collide at your leisurely pace, not effective warp speed.
Similar to warp fields in fiction gravitational waves do this space bending in real life. But their amplitudes are extremely small (10^-21).
Because of the extremely common parlance “drop out of warp” I want to say it implies that the ship keeps moving afterward.
When I run and carry my coffee it is “at warp” but when I stumble and drop it then it is no longer “at warp” but yet it still falls away from the point I dropped it.
But when you drop an abstract there is no implication of movement; when I drop a bad habit it just ceases to exist.
But since the ship still exists after dropping from warp, this is unlikely to be the intended meaning.
Ergo ships drop from warp and that dropping imparts momentum/inertia to ships.
In the "real" world, Alcubierre drives have really interesting (read "devastating") affects on random matter interacting with the warp bubble. The bubble compresses matter in the front and creates micro singularities (which don't necessarily go away when you drop the bubble).
In ST, I'm sure the debris does whatever the writers decide it does. I have no trouble imagining a DS9 episode in which the station gets pelted by warp velocity debris.
I think we've seen enough to safely say it would fall out of FTL in fairly short order.
I don't think we've seen enough to say exactly what its sublight velocity would be - no examples are coming to mind.
Edit: as is often the case when the warp drive comes up, its worth pointing out that ships at warp pretty definitely have momentum/inertia.
It's arguable that it would stop completely, if it survived falling out of the warp field.
The times we've seen ships fall out of warp for one reason or another, they typically slow to a stop, rather than keeping all the inertia like we would expect them to, or stopping instantly like they would if the warp field was doing all the work.
I would think that any object (including the ship) is traveling at a sub-light speed within the warp bubble and therefore would only keep that same velocity when (catastrophically) exiting the warp bubble. Unless by exiting the warp bubble in an uncontrolled manner creates some other force which slows the object somehow.
My understanding is that the warp bubble is moving space around the object (including the ship) rather than accelerating the object to FTL (faster-than-light) speeds, thus we really only have to consider the relative velocities within the warp bubble.
Edit to add: Oh, also, I should add that (IMO) the object cannot continue to travel at FTL speed since it has no warp drive of its own to maintain the warp field.
Disintegrate or just stop and drop out of warp just like the ship does when they disengage warp normally. The ship doesn't travel faster than the speed of light. It just warps space around the ship. That's why they don't have relativistic speeds like the warping of time like time dilation on the ship doesn't happen.