I want to know if heavy elements are distributed roughly evenly throughout the Solar System or if they are (excluding the Sun) concentrated mostly in a particular area, such as the inner part. Would heavy elements be rarer in the Kuiper Belt and outer planets for instance?
Initially, in the protostellar disc, the heavy chemical elements would have been uniformly distributed with radial distance from the Sun, with a mean abundance that is similar to that present in the Sun today (with one or two exceptions like lithium).
But after that there are various processes that act to concentrate or deplete heavier elements, or materials that incorporate those heavy elements. The net outcome is that planets in the inner Solar System have a much higher fraction of heavy elements than the outer bodies in the Solar System. The gas giants Jupiter and Saturn have the lowest fraction of heavy elements.
The most important of segregation mechanism is that in the inner Solar System, the only solids that can form would be those that involve heavier elements like iron, silicon and magnesium. These would combine with oxygen (a quite abundant heavy element) to form silicates, and other minerals. "Lighter" and more volatile solids such as ices (water, carbon dioxide, ammonia, methane) could only form in the cooler conditions further out beyond what is called the snowline (or frost line) at around 3 au from the Sun.
In the inner Solar System, the planets form from the coagulation of this solid material and conditions were too hot for those solid bodies to retain hydrogen and helium gas and thus they are depleted of these elements. In the outer Solar System, bodies could incorporate both rocky and icy solid bodies and then they could also rapidly capture and retain the abundant hydrogen and helium gas that was around them and this would form the bulk of their mass. Thus the giant planets like Jupiter and Saturn have an overall abundance that is quite similar to the original solar nebula (and the Sun itself).
The less massive "ice giants" (Uranus and Neptune) likely formed in an environment where they were unable to capture very much gas before the primordial disk was dissipated, so they still retain an "icy" abundance pattern.
The same is true for a lot of the smaller outer Solar System bodies - they are a mixture of rock and ice and were never large enough to capture a significant amount of hydrogen/helium gas.
Note though, that because Jupiter and Saturn likely have solid cores that incorporate 10-20 Earth-masses worth of solid rocky/icy material, that even though the overall fraction of heavy elements is low, the absolute mass of heavy elements in the cores of the giant planets is probably greater than the sum of what is in the inner Solar System.
Initially yes(1), but actually not anymore.
Initially the chemical composition in the solar nebular was uniform. However a such protoplanetary disc is subject to a radial temperature gradient. Thus the chemical composition of the solid material available for planet formation differs: in the inner solar system only those solids are available which form solids at higher temperatures and at about Jupiter distance even water ice becomes a solid and thus also available for planet formation. Of course, in the later stages, some mixing and migration occurs, but this initial segregation due to the thermal gradient in the protoplanetary disc is still evident today by the different chemical composition of the planets, e.g. the average density at zero pressure (thus taking the pressure and resulting compaction due to size differences out of the equation) of the terrestrial planets decreases from inside to outside. The Earth is a bit the odd planet in this - but fits the image better, if you take the average chemical composition of Earth and Moon combined: it is assumed the Moon formed by the impact of a Mars-sized proto-planet with proto-Earth and as a result the heavy core mostly remained in Earth and many of the lighter-element ejecta formed the Moon. See these values e.g. in this paper by McDonough & Yoshizaki (2021), fig. 4). A similar analysis can be done for the overall solar system. The main feature there will be the higher water content for bodies outward, starting with Jupiter: they are outside the snowline, thus water is already agglomerated in the initial formation process and need not be gained later when the body is large enough to hold liquid or gasous water.
Actually the condensation temperatures of different elements (and even isotopes of the same elements, e.g. here Willacy et al (2015) ) and especially minerals are so well understood that their presence or absence allows to pin-point quite exactly at what temperatures, and using these as proxy for distance from the proto-sun these stones or bodies did form, and from radio-isotopic dating also their exact age (and the time they last were molten).
As summary: on the large scale the solar system is chemically identical, but the small (and partially not-so-small) chemical difference tell us a lot about the history and formation processes.
note 1) In many models one assumes an initially homogenous nebular. But of course there ARE some variations and even the turbulence and turbulent mixing cannot completely remove the spatial differences. Thus one can somewhat try to trace differences back to different parts of the solar nebula and their origins.