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.