How do (and don’t) particles emerge from fields? I am aware of the following field- and particle-like notions:

*

*QFT particle, a unit of excitation in (the Fock space of) a QFT;

*SR field, an extremal $A = A(\mathbf x)$ of a Lorentz-invariant action;

*QM particle, a time-dependent element of a Hilbert space with a position basis $\lvert x(t)\rangle$, with an associated unitary evolution law;

*SR particle, an extremal $\mathbf x(\lambda) = (x^\mu(\lambda))$ of a Lorentz-invariant action;

*eikonal function, a solution of the eikonal equation from optics;

*principal Hamilton’s function, a solution of the Hamilton-Jacobi equation for a Galilean invariant Hamiltonian (not involving $c$);

*CM particle, an extremal $\mathbf x(t) = (x^i(t))$ of a Galilean invariant action (not involving $c$).

The non-relativistic limit (4) → (7) is standard for a spinless massive particle. The same argument gets you (2) → (3) in the sense that it reduces the Klein-Gordon equation to the Schrödinger one. Classical mechanics tells us how (6) and (7) are about the same thing. JWKB is (3) → (6) for spinless particles, and the eikonal approximation is (2) → (5) for a free scalar field in basically the same way, but it’s not $\hbar\to 0$ but $\lambda\to 0$.
There are clearly things missing from the picture. First, there should be some sort of correspondence between (1), (2) and (3) for massive scalar fields. (1) → (2) via functional integration is formally straightforward, but what’s the limit taken exactly? How to do (1) → (3) (preferably in an interacting theory) and what are the limits I don’t know at all—probably with a Schrödinger field as an intermediary step?  One would also want ensure that the “diamond diagram” “closes” to classical mechanics (6, 7) coherently.
Second, there should be a correspondence about (massive) fermions: antisymmetric Fock space/integration over Grassmann variables in (1), the Dirac equation in (2), Pauli equation in (3), something in (7). On one hand, (1) → (2) via the functional integral is again formally straightforward, but it’s not clear what the parameter in the limit is; (2) → (3) is somewhat better.  On the other hand, I’ve heard it said that there are no classical fermions because Grassmann variables classically can’t be anything but zero. Clearly there are: those in the CRT are pretty classical in the uncertainty-relation sense. But the Grassmann variables are zero, which makes one wonder what the stationary-phase approximation in the functional integral even means.
Finally, there’s the business about massless particles. Again stationary phase (1) → (2) works, but in what limit? Whether and in what sense the wavefunction in (3) even exists I don’t understand. (“There’s no photon wavefunction, but there’s this function [the Maxwell field] whose absolute value squared gives you the probability to absorb a photon”, anyone? It is for this reason that didn’t define “position basis” in (3).) Non-relativistic limits of massless particles are bound to be confusing, of course. At least the difference between (5) and (6) makes sense (there’s no $t$ dependence in the former because the photons move too fast), but I’m not aware of any formalization of the ray in geometric optics that would be analogous to (7). Zurek mentions in passing that states with different photon occupation numbers do not decohere, but doesn’t elaborate. Still we somehow compute Compton scattering using massless particles in SR.
A reasonable answer to this question would probably contain a greater percentage of references than usual in order not to get too long. I understand that it screams too broad, but really can’t untangle this blob of confusion in my head. In a sense helping me split it into several follow-ups would also be an answer.
 A: First, let me point out some important thing. There's a definition of particle that unificate, in some way, (1), (3), (4) and (7). This definition is: "a particle is a subspace of the Hilbert space of the theory that furnish an irreducible projective representation of the symmetry group of the theory". We can see more here and here.
Our daily life experience tell us that particles are dimensionless points. When you try to do a kinematics of this particles you are naturally guided to work with trajectories $x(t)$ as you defined in (7). Then we discover that Nature has some laws, and that this trajectories obey some equations that can be understood as a stationary principle.
Then, you may see that some phenomenas are better explained not in terms of particle but in terms of fields, like electromagnetism. When you sit down in a lab and try to find laws for this fields, you discover that this laws are incompatible with our notions of simultaneously events. This lead to SR and our revision of the laws that describes trajectories of particles, as you put in (4).
Now, Quantum Mechanics (QM). When we start to apply our physical ideas to clean systems, i.e. systems with well controlled numbers of degrees of freedom, and small inertia (mass), we are guided to a new way of desciption about Nature. Now, we talk about probabilites of possible outcome after a measurement. We can use QM rules to calculate this probabilities.
Inside the engines of QM, there's a prescription about how you should think in terms of counterfactual reasoning, namely superposition principle. When we try to apply QM on the non-relativistic particles, there's a set of questions as $|x (t)\rangle \langle x(t)|$, that is, what is the probability of the particle being detected at position $x$, $t$ times before the preparation of the system. This is QM of a non-relativistic particle.
Now, QM+SR, with little of math, can show that $|x (\lambda)\rangle \langle x(\lambda)|$ is not an acceptable set of questions, that is, $\langle x(\lambda)|y(\lambda)\rangle$, is not zero for space like separations, specially at equal times (for any referential). All te problems boils down to fact that in SR there's future, past and a time-limbo, i.e. there's pairs of events that can't be assigned by a casual order. There's always a lorentz transformation that invert the order.
But, we can work with a diferent set of questions, namely $|p,\sigma, n\rangle\langle p, \sigma, n|$, where $p$ is the momentum, $\sigma$ is the spin or helicity state, $n$ is the specie of the particle. This questions is acceptable because $|p,...\rangle$'s are orthogonal each other. You may try to construct the $|x\rangle$ by a suitable superposition of this formers states and verify the stataments of the last paragraph. All the problems with the localization of particles boils down to the fact that (using spinless particles for the sake of simplicity),  
$$
|x\rangle = \int \frac{d^3p}{(2p_0)^{1/2}(2\pi)^{3/2}}e^{ipx}|p\rangle$$
Note that the only difference between the non-relativistic case is the $(2p_0)^{1/2}$. This terms spoils the orthogonality.
In terms of creation and annihilation operators this translates to the fact that the creation and annihilation operators of this states do not commute.
$$
[\psi(x),\psi^{\dagger}(y)]\neq 0
$$
Now, to make the locality manifest, we may try to find diferent superpositions of creation and annihilation of this localized state in such a way that they commute for space-like separations. This leads you automatically to a QFT. That is:
$$
\phi(x)=a\psi(x)+b\psi_c^{\dagger}(x)
$$
such that:
$$
[\phi(x),\phi^{\dagger}(x)]=0
$$
Where this $_c$ stands for a charge conjugated particle. You can see that this fields creates states that could be understood as: absorption of particle or emission of a antiparticle. This means that this two process are indistinguishable by the superposition principle.
Because the $p_\mu p^{\mu}=-m^2$ of particles, this fields obeys the Klein-Gordon equations. So, we can construct coherent state of this fields, because they obey a second order differential equation. This states describes Classical fields, i.e. the quantum fluctuations of the fields measurement are well controlled (do not increase with time evolution). But, in non-relativistic QM you can work with this states as well, and then, you may have fields there as well. The difference is that in non-relativistic QM the number of particles commutes with everything in the theory, and is quite impossible to prepare this state.
Now, how to construct classical particles from QFT. Well, to do that this particle nees to have a mass. This mass provide to you a scale $1/m$ of space, the overlapping between the set of questions $|x (t)\rangle \langle x(t)|$ is controlled by this escale. For length scale much moore large that the Compton wavelength this overlapping is negligible and you can consider this questions as acceptables. Then , choosing scale of space and time such that $\hbar$  in front of the mass of the particle is small, there's wave packets with well controlled quantum fluctuations of the observable $x(t)$, i.e. very slow spreading.
Free QFT at boring! How this presents in interacting QFT? Simple, every time that you read particle think in a bound state. A electron could be thought as a bound state construct out of the interactions between the electromagnetic field and the dirac field. This bound state    can break under measurements of small distance physics. This state has large fluctuations in this kind of local o bservables, but for large distance observables $\gg 1/m$ the fluctuations are small, and you can considere the state as intactable after the observation.
Now, constructing wave packets with large width $\gg 1/m$, don't destroys the bound state of the interacting theory. Then, every thing follow straightforward. For massles particles in a completely diferent history.
Massless particles can't be assigned by any question as $|x (t)\rangle \langle x(t)|$, because the mass is zero, and then, $1/m$ is infinite. There's more technical facts that tells you that this particles can only be described by fields in terms of gauge bosons. The typical interaction of abelian gauge bosons with fermions tends to create more coherent states of bosons than localized bound states (particles). Non-abelian theoris I can't tell. In the case of abelian gauge bosons, as photons for example, at high temperature they shows up some non linear behaviour with can be viewed as a particle behaviour maybe (I'm not sure).
A: Here is an experimentalist's answer to the title , for what it is worth.
Your list is actually a list of frameworks where predictive mathematical models of observations have been constructed.
The frameworks depend on the ranges of the variables available for measurement, space-time and energy-momentum. As one goes to smaller and smaller ranges different formulations succeed in describing observations.When one hits the Heisenberg Uncertainty ranges , quantum mechanics becomes necessary to model the observations with a predictive theory.
Let me illustrate: If you watch a turbulent sea, you can follow a wave up to the shore and give it an identity as a specific crest. Is it a particle? Why is it specific? Because it is conserving energy and momentum in space-time. One does not call it a particle because it does not have a specific (x,y,z,t) and mass, which is what we call particles in everyday language. A soliton wave though in water, behaves more as a  particle. 

In analogy one can see the emergence of particles from QFT as a soliton type  form conserving not only energy and momentum but also a number of quantum numbers , which for dimensions larger than the Heisenberg constraints are localized in (x,y,z,t) and (E,p_x,p_y,p_z) within measurement accuracy into points in space and specific masses .

My answer to a related question here is also analogous.
A: If we look as wave vs particle, the wave wavefunction is an eigenvector of the momentum, a particle wavefunction is an eigenvector  of the position. (1) is more wave-like than particle-like, a particle-like wavefunction can be constructed by superposition of infinite $|p(t)>$ wavefunctions "a la Fourier".
(1)->(5) is going to classical limit, like (3)->(4). (4)->(7) is taking the newtonian limit on mechanics. (1) and (3) are different bassis for the same. (2) expectation value under appropiate boundary, initial and final conditions -> (6) in the classical limit.
A: Particles are simply high momentum wave states interacting weakly with matter, i.e. their "existence" is observer dependent. Trying to derive them from some form of free field equation is therefor useless and so is the assumption that they are a general phenomenon. They are a highly likely phenomenon for energies that are much higher than the typical em interaction energy of a charged "particle" or high energy photon in a solid, liquid or gas, which can be as little as approx. 1 eV (photo effect) or even much less if we include the phonon excitations measurable with cryogenic superconducting bolometers. The probability of being almost exactly in the rest system of any one high energy plane wave is low and the probability of being in the rest system of all possible high energy plane waves is zero and therefor these localized particle solutions have to emerge naturally. Mott noticed and correctly derived them in 1929 in his paper about the emergence of $\alpha$-particle tracks from wave mechanics. That this has gone widely unnoticed in the teaching of quantum mechanics is very unfortunate, though. 
