I think the crux of your question stems from the apparent pattern in the observed gauge groups appearing in the standard model. In particular, we see a $U(1)$, then $SU(2)$, then $SU(3)$, so if we follow the pattern we might guess this is just the beginning of an infinite series of gauge groups appearing, so the next would be $SU(4)$ (note this pattern isn't perfect, i.e. one would think we should use $SU(1)$, which is actually just the trivial finite group of one element). First I'll say that recognizing patterns and asking if there is an underlying explanation is absolutely essential to advancing physics from a theoretical perspective. And often the most profound breakthroughs come from seemingly trivial observations (the discovery of the different quarks seemed to follow a similar pattern: they had two, then it looked like 3 worked better, then they needed 4, and so on). So all that is just in support of the question, and also to refute the argument that the answer is "that is just the way nature is."
So once you have recognized a pattern, you should start asking whether the pattern solves existing problems with the your current understanding of the system. In the case of quarks, the two quark model did a good job explaining the pion particles that showed up at low energies. However, as more particles were discovered, it looked like they were arranging themselves into groups of $8$ or $10$ rather than groups of $3$. The explanation seemed to be that there was an underlying $SU(3)$ symmetry (not to be confused with the $SU(3)$ color gauge symmetry!), which required $3$ quarks, instead of the previous model based on $SU(2)$ symmetry with $2$ quarks. In fact, after thinking about how particles behaved under the electroweak interaction, they further realized a fourth quark was needed (although the corresponding $SU(4)$ symmetry you might guess is present is actually not, since the charm quark is too heavy to be considered on the same ground as the lighter three). Of course, now we know that there are $6$ quarks, and still people like to speculate whether there could be more.
So back to the original question of whether extending the pattern of the observed gauge groups solves any problems with the standard model. As far as I know, adding an additional $SU(4)$ symmetry doesn't do much other than add more particles that we haven't seen. So those prospects do not look good. However, a similar question related to the structure of gauge groups in the standard model is whether it arises from a grand unified theory (GUT), where the standard model gauge group appears as a subgroup of a larger gauge group. It turns out the smallest simple group that contains the standard model's $SU(3)\times SU(2)\times U(1)$ is $SU(5)$, and there are a number of interesting ways how the particles in the standard model arrange themselves into nice representations under $SU(5)$. This unification solves an interesting problem about how the gauge couplings in the standard model all seem to run to the same value at high energies, which would be an extraordinary coincidence in the absence of a GUT explanation. In this case, the simplest $SU(5)$ models don't seem compatible with data, but extensions involving $SO(10)$ or supersymmetry (as well as a host of other things) still look promising.
In fact, $SU(4)$ can show up as a subgroup of $SO(10)$, and so $SU(4)$ may play an important role in this GUT. I believe in this version of grand unification, lepton number plays the role of the fourth color. So for example, the three colors of up quarks and the neutrino arrange into a four color multiplet of $SU(4)$, and the three colors of down quarks combine with the electron to give another $SU(4)$ multiplet, which is kind of neat!
Anyway, I hope this gives you some intuition about how and why an $SU(4)$ gauge group could arise.
Pauli's exclusion principle is not a force, but rather a constraint that has the effect of changing how other forces manifest.
The relevant force in this case is still just the electromagnetic force (what other ones could be involved, by process of elimination?). But because of how that quantum mechanics alters the informational attributes of mechanics, the quantity of "electromagnetic force" between two particles must, just like position, be described as a probability distribution, viz. a "force wave function", only this probability distribution is not over points in space but over possible values of the electric force felt by the electrons in the materials.
And what Pauli does is alter that probability distribution. Thus it does contribute to determining what the final force you feel and can exert with your hand is. But a nontrivial distribution for the force is only present to begin with by virtue of the fact that there is an electromagnetic force source available.
ADD: That said, though I haven't worked through the details, it might - so don't trust me on this - be that you could write that altered distribution as though it were generated by a "phantom force" operator acting on an otherwise non-Pauli "counterfactual" state (think combining electron states as though they were bosons) and, in this sense, you could think of it as a "force" in a manner analogous to how that centrifugal, Coriolis, etc. effects are understood in Newtonian mechanics. In any case, though, the counterfactual state cannot be correct because it will yield the wrong probabilities for other physical parameters (though maybe you can also distort their operators in a suitably complementary manner to even that out, so you are in effect "shifting a reference frame" on the operator space. Still, though, the main point holds that this is a mathematical trick.).
Best Answer
The notion of "conservative forces" is not in any way fundamental. What's fundamental is that we're able to assign a number to the state of a system, and that number is conserved.
The relatively uninteresting notion of a "conservative force" can be applied only to a force that can be expressed as a vector field that depends only on position. That means it's meaningful for Newtonian gravity and for electrostatics, but not for any other force that could be considered fundamental. Re the nuclear forces, see Do strong and weak interactions have classical force fields as their limits? .
General relativity has local conservation of energy-momentum, which is expressed by the fact that the stress-energy tensor has a zero divergence. A mass-energy scalar or energy-momentum vector isn't something that can be defined globally in GR for an arbitrary spacetime.
WP says (Gugg, where was the link from?):
The first sentence is wrong, because Mercury's anomalous precession can be described in terms of a test particle moving in a Schwarzschild metric. The Schwarzschild metric has a timelike Killing vector, so there is a conserved energy-momentum vector for test particles.
The second sentence is also misleading, since it doesn't make the global/local distinction. What's conserved locally isn't a pseudotensor, it's a tensor (the energy-momentum vector). Globally, there are various pseudotensors that can be defined, and the fact that they're pseudotensors rather than tensors means that they're fundamentally not well-defined quantities -- they require some specially chosen system of coordinates.