Do the planets, asteroid belt, kuiper belt, and scattered disc lie on the same plane? If not, do any of the four groups in the title lie on the same plane as other members of the group? E.g. does the asteroid belt lie on the same plane as the planets do?
They are not exactly in the same plane, but roughly are.

For example this graph representing the centaurs, the Kuipier belt and scattered disc objects show that most objects are around the same plane, with few objects with more than 30 degrees of inclination:



This is the inclination of the asteroids:



Again, few have more than 30 degrees of inclination.

The planets are also all roughly in the same plane. Lets take the ecliptic as a reference (which is almost the same thing as Earth orbital's plane):


The Sun's equator has an inclination of 7.25 degrees.
Mercury has an inclination of 7.00 degrees.
Venus has an inclination of 3.39 degrees.
Mars has an inclination of 1.85 degrees.
Vesta has an inclination of 7.13 degrees.
Juno has an inclination of 12.98 degrees.
Ceres has an inclination of 10.59 degrees.
Pallas has an inclination of 34.84 degrees.
Interamnia has an inclination of 17.29 degrees.
Hygiea has an inclination of 3.84 degrees.
Jupiter has an inclination of 1.30 degrees.
Saturn has an inclination of 2.48 degrees.
Uranus has an inclination of 0.77 degrees.
Neptune has an inclination of 1.76 degrees.
Orcus has an inclination of 20.57 degrees.
Pluto has an inclination of 17.15 degrees.
Ixion has an inclination of 19.58 degrees.
Salacia has an inlination of 23.94 degrees.
Varuna has an inclination of 17.2 degrees.
Haumea has an inclination of 28.19 degrees.
Quaoar has an inlination of 7.99 degrees.
1992 QB1 has an inclination of 2.19 degrees.
Makemake has an inclination of 29.00 degrees.
Eris has an inclination of 44.04 degrees.
Sedna has an inclination of 11.92 degrees.


Most of those bodies have a low inclination to the ecliptic, but Pallas is an interesting exception. I don't know why Pallas has such a high inclination but I would bet on a Kozai mechanism due to Jupiter's influence.

Plutinos and Kuipier belt objects are influenced by Neptune, and part by the Kozai mechanism.

Eris is a special and curious case, it is too far from Neptune, although there is still some weak gravitational Neptune's influence there, but I don't know why its inclination is so high. This contrasts with Sedna, which have a more reasonably lower inclination.

Further, the Oort cloud is expected to be full of bodies of essentially unrestricted inclinations and possibly very excentric orbits, as a result of gravitational interactions with passing stars on the past. By the way, a passing star in a distant past might be an explanation for Eris orbit and inclination.

Note: All of the data here, as well as both of the images, I found on wikipedia.