The properties of protons, neutrons, and electrons that make them unique/distinct from each other (mass, electric charge) arise from the composition of each out of smaller particles, which each are/carry/act as the respective mass + charge + the other properties.

Protons and neutrons are composite, made out of two up and one down respectively two down and one up quark. Plus gluons holding them together. There are also four other quarks, bottom, top, charm, and strange. And of course one antiquark for each of the six quarks. There is a huge number of particles made out of quarks, called hadrons. Hadrons are either baryons like the proton and neutron made out of three quarks, or mesons made out of one quark and one anti quark. There are also exotic things like tetraquarks.

Electrons are, as far as we know, fundamental and not made out of other particles. The same goes for the muon, the tau, and the three accompanying neutrinos. There is again an antiparticle for each particle. This group is called leptons.

The properties of composite particles are determined by their constituents, but not in a trivial way. The mass for example is usually bigger than the mass of the constituents because the binding energy contributes to the mass.

Is there any theory as to what this "looks" like? Or is the best we can do "it's a bunch of these things mashed together and the only way to see them individually is to bash them together until they break"?

Quantum chromo dynamics is the theory of quarks and gluons.

If the strong and weak forces are particles, does that mean they're 1: literally everywhere, not necessarily stuck to any larger particle and 2: like glue?

The electromagnetic, the strong, and the weak interaction are mediated by their respective bosons, we also suspect it for gravity. You can observe the bosons on their own, they are not like springs and rubber bands connecting particles between wich they mediated forces. Actually there are not really any photons bouncing back and forth between two electrons pushing them away from each other due to their like charges. But I can not offer any good model, that is something I never managed to really understand.

I'm also confused about the relationship between gravity and mass, given that the higgs is stated as corresponding to mass, mass is traditionally thought of as what gravity acts upon, but the wikipedia chart states that gravity acts upon all particles.

Most mass comes from [binding] energy, the Higgs mechanism contributes only a small bit. The Higgs boson has nothing to to with that at all, it is just an excitation in the Higgs field. Gravity acts on energy. As far as I can tell mass is just an abstraction. If you put massless photons into a mirror box to bounce around, they add energy to the box which makes the box harder to move, i.e. you have to push against the photons hitting the wall you are pushing on. As a convenient abstraction we say the box got heavier, it has more mass, but there is actually nothing fundamentally heavy in the box, the photons have no mass, only their energy and momentum with which they hit the wall making it harder for you to push it.

I am not a physicist, take all this with grains of salt.