1. How and why do the shuttlecraft leave an antimatter trail ("Metamorphosis")?
There's no actual mention of a trail - the heroes seem to merely expect a spot occurrence, and fail to find one.
OTOH, that "antimatter residue" would
not be the same thing as "residual antimatter" is a good idea in theory, but the very next phrase casts doubt there: Scotty argues that even if the shuttle were towed away, there would be "residual matter floating around". Again, no mention of a trail, though - so if a shuttle leaks (anti)matter at all flight modes, why doesn't it do so consistently, so that it would create a trail? Indeed, what about the Companion's antics caused the shuttle to fail to create "antimatter residue" or "residual matter floating around" if those were expected?
I guess the simplest way out would be to assume that the heroes were on a dark mood that day, and looking for signs of
destruction or at least
damage specifically - supposedly, nothing else would explain Kirk's failure to make contact. There was this asteroid belt there, so the likely failure mode would be a collision, resulting in leakage and flotsam.
Indeed, at the conclusion of the scene, Scotty says that what they have established is that "it didn't wreck". Antimatter residue could be an issue specifically related to wrecking.
2. Does the Enterprise leave an antimatter trail on Impulse or even Warp?
In "Pirates of Orion", it's said that the pirate ship creating a "unique" trail of radioactive waste was helpful in tracking her down - the inference might be that trails are common, even confusingly so, but uniqueness helped sort out the bad guys from the innocent passersby. OTOH, warp vs. impulse is not specified: the heroes give warp chase, but only for a brief while, and well after the villains departed. As elsewhere in Trek warp tracking (or tracking across interstellar distances) is exclusively via "warp signatures" which seem to be point phenomena rather than trails, we might argue that the Orions here were moving at impulse.
We probably have to apply the "warp factors are slower insystem" clause to make that work. But the action does seem to involve a star system, as it involves an asteroid "belt" (rather than, say, a "field")...
Timo Saloniemi
