Does the existence of exchange particles imply that that the 4 fundamental forces are delivered in discrete packets instead of continuously? If exchange particles transfer the fundamental forces and these particles takes some amount of time to transfer this force does this mean there is a rate of force?
(Side question: if two oppositely charged particles are infinitely far apart will they be attracted to each other as the photons would take infinitely long to travel between the particles?)
 A: Here's a model you might have in mind given the language that forces are carried by exchange particles. Imagine two electrons separated by a distance $R$. To exchange the electric force, photons must travel from one electron to the other. Since there is a discrete number of photons, you would expect that on sufficiently small time scales you could see Poisson (random) fluctuations in the number photons arriving per unit time. Therefore you would expect the force of the electron to fluctuate by some amount on small time scales.
This model is not correct. The photons that exchange the electrical repulsion are virtual photons. This means you shouldn't think of them as real particles, but as a kind of disturbance of the electric field. They are really a mathematical convenience for describing terms in a perturbative expansion. Matt Strassler has a very good write up of what a virtual particle is for a non-technical audience here: https://profmattstrassler.com/articles-and-posts/particle-physics-basics/virtual-particles-what-are-they/ (Although also see this SE answer for a discussion about where Strassler's explanation may differ from the standard definition: physics.stackexchange.com/q/230113/50583)
Additionally, because of quantum mechanics, there is not a single discrete process that occurs where a photon is exchanged between the two electrons. What really happens is that there is a certain probability for the electrons to move apart by some amount because of their electrical repulsion. This probability can be computed as a series of terms, one term involving the exchange of a single virtual photon, another term involving the exchange of two virtual photons, and so on. The leading order approximation (in the limit that electrons move non-relativistically and that the coupling of the electrons to the photons is small, which it is in reality) is that the electrons repel each other according to the usual Coulomb force law.
A: Not sure what you mean by “rate of force”, but we know that the electromagnetic force, the weak force and the strong force are delivered in discrete (but very small) packets or “quanta”. The jury is still out on whether this applies to gravity as well.
Also, all four fundamental forces (including gravity this time) have a maximum speed of transmission, which is the speed of light in a vacuum.
Finally, you cannot physically have particles that are “infinitely far apart”, but you can ask what happens as the distance between the particles becomes greater and greater. The electromagnetic force between two charged particles decreases as the square of the distance between them, so in the limit, as the distance becomes greater, the force tends to zero.
A: 
If exchange particles transfer the fundamental forces and these particles takes some amount of time to transfer this force does this mean there is a rate of force?

Exchange particles are defined within Quantum Field Theories that use the Feynman diagram expansion to calculate the interactions between particles.
All exchanges within a Feynman diagram  can represent a force, the dp/dt , assigned to a virtual exchange particle, and all particles in a theory can be virtual exchange particles.In the standard model  of particle physics these can be any of the particles in the table.

See this  photon electron scattering feynman diagram where the exchange virtual particle has the quantum numbers of the electron and is represented by a propagator with the mass of the electron, but its four vector is off mass shell, as in all particle exchanged. (page 2 for definition of virtual)
In a nuclear physics field theory the exchanged particle can be a pi0 or a rho meson, nothing to do with fundaments.
What are called fundamental forces are the three SU(3)SU(2)xU(1) symmetries of the particle physics , where the gauge bosons (the fourth column in the table) correspond to each  special unitary group of the theory.
Force at the level of particles is the dp/dt transferred at the vertices.

(Side question: if two oppositely charged particles are infinitely far apart will they be attracted to each other as the photons would take infinitely long to travel between the particles?)

In mainstream physics the theoretical expressions are such that Lorentz transformations hold and nothing can go faster than the velocity of light in vacuum, c. So if one went into the trouble to write the attraction of opposite charges at infinity using quantum field theory, on would find mathematically that there would still be an attractions, as infinity is never reached anyway.
A: Imagine empty space being filled with virtual particles that are all over the space and go back and forth in time. Or, equivalently, have all energies and 3-momenta, independently from one another, not obeying the relativistic energy-momentum relation ($E^2=p^2+m^2$). I say equivalently because time and energy are  both the time component of a 4-vector, and position and 3-momentum the space components, and the position 4-vector and energy-momentum 4-vector are complementary.
Now, real particles, when interacting, couple to this virtual field, which delivers the right energies, dependent upon asymptotic boundary conditions. The charge of the particle determines to which fields it couples. Of course the virtual fields couple to each other as well, giving rise to higher order interactions. Leading to rather complex looking integrals corresponding to higher order Feynman diagrams. Particles can couple directly to quantum bubbles, closed loops in Feynman diagrams, which causes, for example, anomalous magnetic moments.
So in a first order process, the virtual photon in between, delivers the right package of energy and momentum to both electrons (the deltas in the integral corresponding to the Feynman diagram), just picking the right values from the wide, almost infinite range of both they possess. When the energy-momentum load is delivered, the virtual photon dives back into the vacuum, waiting for next charges to come along.
