Well the Kv is rpm-per-volt (no load) so you can figure this out - this will then give you the torque-per-amp directly as they are related (hint, convert to SI units and radians/second and the constants are the same)
You have to measure the winding resistance. The max current is the current rating to avoid overheating, not related to your parameters. The MoI you'll have to measure again, although you can estimate this from the mass and dimensions of the armature. Damping I've no idea, there's a lot of friction of various kinds. Inductance of a motor is not well defined since the magnetic circuit is not constant, saturation is likely to be happening, and back-EMF will complicate things - again you can try measuring this with an inductance bridge, you won't be able to deduce it.
Yes every DC motor that's different in dimensions will be different, and many will be the same design but differently wound and so electrically different.