The problem of self-force on point charges Allow me to preface this by stating that I am a high school student interested in physics and self-studying using a variety of resources, both on- and off-line, primarily GSU's HyperPhysics website, Halliday & Resnick's Fundamentals of Physics, Taylor's Classical Mechanics, and ultimately the Feynman lectures (mirrored by Caltech). Hopefully this gives somewhat of a feel of my level of physics understanding so as to avoid any answers that fly far above my head.
As I've understood from previous reading of electromagnetism (for example, in Halliday), a point charge is not affected by its own electromagnetic field. Unfortunately, as I recently read in the Feynman lecture on electromagnetism, this appears to not be so:

For those purists who know more (the professors who happen to be reading this), we should add that when we say that (28.3) is a complete expression of the knowledge of electrodynamics, we are not being entirely accurate. There was a problem that was not quite solved at the end of the 19th century. When we try to calculate the field from all the charges including the charge itself that we want the field to act on, we get into trouble trying to find the distance, for example, of a charge from itself, and dividing something by that distance, which is zero. The problem of how to handle the part of this field which is generated by the very charge on which we want the field to act is not yet solved today. So we leave it there; we do not have a complete solution to that puzzle yet, and so we shall avoid the puzzle for as long as we can.

At first I figured I must've misunderstood, but upon rereading, it's clear Feynman states the that electromagnetic field due to a point charge does, in fact, influence said charge; I inferred this "self-force" must be somewhat negligible for Halliday to assert otherwise. What stood out to me was that Feynman states this problem had not yet been solved.
I suppose my first major question is simply, has this problem been solved yet? After a bit of research I came across the Abraham-Lorentz force which appears to refer exactly to this "problem of self-force". As the article states the formula is entirely in the domain of classical physics and a quick Google search indicates it was derived by Abraham and Lorentz in 1903-4, why is it that Feynman state the problem was still unsolved in 1963? Has it been solved in the classical case but not in QED? 
Lastly, despite the Wikipedia article somewhat addressing the topic, is this problem of self-force present with other forces (e.g. gravity)? I believe it does state that standard renormalization methods fail in the case of GR and thus the problem is still present classically, though it does mention that non-classical theories of gravity purportedly solve the issue. Why is there not a similar Abraham-Lorentz-esque force possible in GR -- is there an underlying fundamental reason? Due to the relative weakness of gravity, can these self-force effects be ignored safely in practice?
I apologize for the long post size and appreciate any help I can receive. I only hope my post isn't too broad or vague!
 A: The electromagnetic self-force problem has been solved recently, see here; the gravitational self-force problem has also been solved recently, see this article.
A: 
I suppose my first major question is simply, has this problem been solved yet? After a bit of research I came across the Abraham-Lorentz force which appears to refer exactly to this "problem of self-force". As the article states the formula is entirely in the domain of classical physics and a quick Google search indicates it was derived by Abraham and Lorentz in 1903-4, why is it that Feynman state the problem was still unsolved in 1963? Has it been solved in the classical case but not in QED?

This is still only a theoretical problem, as a measurement of the expected self-force needs to be very sensitive and was never accomplished. Theoretically, self-force can be said to be described satisfactorily (and even there, only approximately) only for rigid charged spheres. For point particles, the common notion of self-force (Lorentz-Abraham-Dirac) is basically inconsistent (with basic laws of mechanics) and can be regarded as unnecessary - for point particles there exist consistent theories like Frenkel's theory or Feynman-Wheeler theory (with or without the absorber condition) and their variations without self-force (there are other works free of self-force too).
J. Frenkel, Zur Elektrodynamik punktfoermiger Elektronen, Zeits. f. Phys., 32, (1925), p. 518-534.
http://dx.doi.org/10.1007/BF01331692
J. A. Wheeler, R. P. Feynman, Classical Electrodynamics in Terms of Direct
Interparticle Interaction, Rev. Mod. Phys., 21, 3, (1949), p. 425-433.
http://dx.doi.org/10.1103/RevModPhys.21.425
A: I stumbled across this question and wanted to reply to one particular part of it:

Lastly, despite the Wikipedia article somewhat addressing the topic, is this problem of self-force present with other forces (e.g. gravity)? I believe it does state that standard renormalization methods fail in the case of GR and thus the problem is still present classically, though it does mention that non-classical theories of gravity purportedly solve the issue. Why is there not a similar Abraham-Lorentz-esque force possible in GR -- is there an underlying fundamental reason? Due to the relative weakness of gravity, can these self-force effects be ignored safely in practice?

The answer is that there is in fact a similar Abraham-Lorentz force in GR.  In most circumstances, it can be ignored.  However, over the past ten years or so, the construction of the LIGO experiment (and its successor) spurred an interest in understanding the details of how massive objects, like stars or black holes spiral in to other black holes.  
As these objects orbit around large black holes, they will emit gravitational waves (the things that LIGO is built to detect.)  These waves carry energy away from the system, and so (if you go through the math carefully) the star must get even closer to the black hole, which causes it to emit even more gravitational waves, which causes it to spiral in even faster, until finally it plunges in.  We've never detected the gravitational waves from this sort of system directly, but we do have all sorts of indirect evidence that convince us that they're being emitted and that they cause tightly orbiting systems to lose energy.  
Still, we'd like to actually be able to detect these waves.  The problem is that there's a lot of noise in gravitational-wave experiments, and although the physicists at LIGO are going to heroic efforts to minimize it, the signal-to-noise ratio that they'll be looking at is still pretty low.  To make it easier to pick these signals out of the noise, one of the ideas in play is to use "templates" to find the signals.  These would be pre-calculated signals that you'd be specifically searching for among the noise, which would make them easier to find (think about how easy it is for your brain to pick up your own name, a particularly familiar "signal", when it's mentioned amid the hubbub at a crowded party.)  But to do this, we would need to know the precise details of the self-force on these stars/black holes as they plunge into the central black hole, since the precise trajectory affects the amplitude and the phase of the wave at any particular time.  Thus, the need to describe the self-force.
An interesting twist here is that electromagnetic self-force has (to the best of my knowledge) never been experimentally observed in the classical regime.1 This means that there's a very real possibility that the gravitational self-force may actually be observed first, via its effect on gravitational wave forms—despite the fact that it is many, many times weaker and was conceived of many years later.

1 I welcome corrections on this point.
A: I'm not sure if this problem was ever solved in classical electrodynamics.  
However, it is (somewhat) solved in quantum field theory electrodynamics (QED).  In QED, self-interaction has noticeable effects on quantities such as the observed mass of a particle.  Furthermore, the self-interaction effects create infinities in the theoretical predictions for such quantities (which is why I said "somewhat" above).  But, these infinities can be cancelled out for any observable (such as energy or mass, etc...)  This process of cancelling out the infinities is known as re-normalization.  
To get a sense of how it works, imagine that your theory predicted the energy of a particle to be something like $$E_\textrm{theoretical} = \lim_{\lambda\to \infty} (\log\lambda + E_\textrm{finite})$$ where $\lambda$ represents the part of our calculation that becomes infinite.  For example, if an integral diverges we can take set the upper bound of the integral to be a variable (such as $\lambda$) and then at the end take the limit as $\lambda$ goes to infinity.  Methods such as these are called "regularization" (i.e. a way of rewriting the equation such that the divergent part of the calculation is contained within a single term). 
Now in this limit, the total energy will be infinite. However, in the lab we can only measure changes in energy (that is, we need a reference point). So, let us then choose a reference point such that $E_\mathrm{0,finite}=0$.  In that case, we subtract the reference point from the theoretical energy to get $$\Delta E_\textrm{observed} = \lim_{\lambda\to\infty} (\log{\lambda} + E_\textrm{finite} - \log{\lambda} - 0) = E_\textrm{finite}$$ and all is well.  This last step is called re-normalization.
