Not without interacting with a 3rd body, such as Jupiter.

Or, apparently, if it crosses an event horizon, although I've never properly understood how that works. (Explanations I've found typically just say: "All paths from a black hole are toward the black hole."

Most things flying towards a black hole don't cross the event horizon thanks to angular momentum.

Stars tend to get torn apart, go into the accretion disk, and then eventually fall in. But we don't have a big black hole in the solar system.

For present purposes the Newtonian approximation suffices, and in Newtonian gravity there are no black holes and no event horizons.

Captured into a stable orbit, sure, but it will never have an eccentricity of 0 (circular). The limit can approach zero, but will never be equal to it. Even the planets don’t have a circular orbit FWIW.

An interstellar object cannot be captured into a stable orbit unless some other force (such as Jupiter's gravity) acts on it.

Orbits are time-symmetric, so an interstellar object being captured into a stable elliptical orbit would be equivalent to an object in a stable elliptical orbit escaping the Solar System (again, ignoring other influences).

Is that true of comets as well? It seems conceivable to me that the sun could spin it around such that the comet's off-gassing could reduce its speed and allow it to be captured. Obviously unlikely, and highly dependent on conditions (in particular incoming velocity) but possible.

If you're ignoring all other influences then there is no such thing as interstellar objects. By definition, every object in a two body system are gravationally bound to each other.

This is so wrong. Objects can have enough energy that will become arbitrary far apart as time passes. In no sense are the objects bound together in that case.