Are particles merely "relatively stable" patterns that can appear on their respective fields? With quantum field theory, particles are seen as excitation on various fields. Am I mistaken to think that then particles merely refer to "relatively stable" patterns that can appear on these fields? I assume the answer is yes, and in that case I have a couple questions: are interactions depicted in Feynman diagrams approximations of what actually happens in the involved fields? And is that why individual nucleons can't be perceived as being "neatly compartmentalized" inside the nucleus, if that even is true, in that if you were able to precisely take a look at the field, you wouldn't necessarily be able to recognize individual patterns corresponding to each individual nucleons?
 A: 
Am I mistaken to think that then particles merely refer to "relatively stable" patterns that can appear on these fields?

In the field theoretical standard model of particle physics, the particles in the table are assumed to exist axiomatically,as point particles with their individuality assured by the quantum numbers and mass. There is nothing "relative" in their presumed existence. The field theory assumes that for each particle in the table exists a field, an electron field , a neutrino field etc. and creation and annihilation differential operators act on these fields to create and destroy the particles. That is the way they are used in the Feynman diagrams.

if you were able to precisely take a look at the field,

Does it have a meaning to "precisely look at the coordinate  (x,y,z,t), there is an infinite number of them? The fields are like that to the creation and annihilation operators.
A: There is no such thing as "looking at a quantum field", and particles aren't just "relatively stable patterns". You're thinking about this with a classical intuition (that there are things and that they have unambiguous properties and that you can look at them), but classical intuition does not apply in the quantum realm, and indeed intuition as such is a hard tool to master in the context of quantum field theory - we need to retrain our intuition to conform to what the theory says, not try to twist the theory into fitting the classical world in which our intuition was formed.
Fields - both classical and quantum - should not necessarily be imbued with the idea that they are some sort of substance we could look at, see this question and its answers for a longer discussion of the ontology of fields. Physics, especially quantum physics, provides a mathematical model of the world that allows us to predict what we will observe, but it does not necessarily select a specific ontology - no unique idea of "what there really is", whatever that means. There is nothing observable about "the electron field" except electrons and positrons, we cannot "look" at this field in any other way than the particles and processes we associate with it.
It is, as far as we know (cf. searches for proton decay), a fact that an isolated proton is infinitely stable, and so is a single photon that doesn't have anything else to interact with, or a single electron. There's nothing relative about this, and nothing approximate. Some particles are stable, others aren't, but this has nothing to do with them being "excitations" - insofar as it is meaningful to say that particles are excitations of fields, all particles are such excitations, stable or not. The impossibility of separating a nucleon like the proton into neat smaller constituents is due to the strongly interacting nature of quantum chromodynamics, which makes the perturbative approach in which the particles we associated with free fields are a good approximation impossible. See this question and its linked questions for more discussion of the internal structure of hadrons.
The problem here is what it actually means to say that a particle is "an excitation of a field". All physicists agree about the technical sense - the modes of a free field become creation/annihilation operators during quantization and then construct particle states - but there is no necessary implication between that technical sense and the vague idea of a particle as some sort of wave in a material field that seems to be implied by your question. See also this question and its answers for a longer discussion of the sense in which quantum fields "oscillate" or get "excited". The problem is, again, imbuing a formal description with intuitve ontological weight it does not actually  carry within itself.
