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.