As we all know, both electrons and protons have this small electric charge 1.6x10-19C or 1e. Where did they get this charge in the first place? Is it eternal or can you neutralize it - like neutral proton charges to get cold fusion?
Standard model of elementary particles. The proton’s charge comes from having two up quarks and one down quark. An up quark has a charge of +2/3 and a down quark has a charge of -1/3. Two ups and a down make for a charge of +1. (A neutron has two downs and one up - do the math.) An electron is a lepton (as opposed to a quark) and has a charge of -1.
Where do the elementary particles get their charge? What is electric charge, really beyond their force effects? I don’t know, maybe it’s turtles all the way down.
Does anyone know why the down quark has a charge of -1/3 and the up has 2/3? That is, why aren’t they equal but opposite?
You cannot neutralize the charge on anything, proton, electron, or otherwise, except by putting it together with something else charged. Most typically, this is done by putting protons together with electrons, but when you do this, you still have protons and electrons, each still with charge. Charge is absolutely conserved in every situation known to exist or speculated in any conceivable extension to known physics. And so far as anyone can tell, cold fusion does not exist, and is only a lie to separate gullible investors from their money.
Why should they be? They’re different objects, and not anti-particles.
nobody knows, therefore god must exist. Is that what you’re getting at?
No, no, no. The logical conclusion is Don’t Know, therefore Leprechauns.
Similarly, Contradiction implies Unicorns. This may well be a bi-implication, but nobody’s gotten it drunk enough.
I am not a physicist, but I don’t think there is any serious scientific answer to your question. This is because the only way we have of learning about things like protons and quarks is through creating mathematical models of their behavior and running experiments to validate the models in reality. But of course the key word there is “behavior”. Since we can only test how these things interact with other things, we are left with properties like electric charge, color charge, etc… where we can describe very accurately the behavior of but have no idea what they are. So almost by definition, science can’t tell us the ontology of the fundamental entities the models are made of.
Static cling. God forgot the fabric softener.
They’re not opposites – that is, antiparticles of each other – else they’d annihilate instead of combining into protons and neutrons. They do have antiparticles, though. They’re named, unimaginatively enough (probably by the same prosaic plodders who renamed the truth and beauty quarks to top and bottom) antiquarks, and have charges +1/3 and -2/3.
Depending on your frame of reference, those terms can be quite romantic - or so I’ve heard. 
Physicists are kinky?
Charge is a property of the fundamental particles, not some kind of separate substance or something. As such, it’s not something you can add or take away. When we think of adding or removing electric charge from a macroscopic object, really we’re talking about adding or removing more charged particles.
As far as why electrons and quarks and whatnot have the amount of charge they do, so far as I know this is an open question. But I should maybe clarify one point, which is that the fact that an electron has a charge of 1.60210^-19 Coulombs is really just a matter of how we defined or units. There’s nothing fundamental about the number 1.60210^-19. However, we can relate this number to other numbers that don’t have units – in particular the charge on an electron is related to the fine structure constant, which is approximately 1/137 (note the lack of units). It’s numbers like that which are really fundamental.
Why these dimensionless (that is, unit-less) numbers take the values they do is an open question so far as I know. Maybe that’s just the way the world is, with no rhyme or reason… maybe there are other worlds (that is, universes) with different values for these numbers, and this just happens to be the one that works out well for human life to arise… maybe there’s some underlying theory of everything that will explain why the value had to be what it is… who the hell knows?
You may not be a physicist, but you sure know how to think like one. Well done.
I can tell you that iophobon’s reasoning here is not representative of the physics community. It is correct that we can only test how things interact with other things – indeed, the current understanding of particle physics is based on interactions between particles and fields. But the belief that this therefore bars physics from deductions or inferences about the fundamental character of the entities we study is almost entirely antithetical to the entire purpose of physics. In particular, from hanging around particle physicists, I can tell you that the general opinion is they all want to find something outside the Standard Model precisely because it has (at last count) 19 numerical constants that must be experimentally measured and inserted into the theory (as well as a few other outstanding issues, like gravity). The hope is that a new theory of particle physics will provide a way to derive some of those constants from new physical principles, which is the first step towards explaining their origin.
To bring this back to the OP, some of those 19 constants are electric charges, and there is as yet no good explanation for why the particles have the charges they do.
Are you calling God a homo?
Electric charge (i.e., the Fine Structure Constant) is only one of the free parameters, though there are others which describe other properties which may be considered other forms of charge (how strongly various particles interact with the Weak Force or the Color Force, say). Or, alternately (and equivalently), you can consider those constants to be a measure of the strengths of the forces themselves.
I’m not sure how this opposes anything I said. A new theory of particle physics will work the same way as previous ones: by testing model predictions against reality. I suppose people are welcome to make guesses about the nature of the entities and why they are the way they are but why would they get labeled as anything other than guesses? If you thought I was suggesting we could never have a more complete understanding of the pieces of current models then I agree with you as that is not my position.
Really, the main point I wanted to make is I think people far too easily jump into the trap of thinking that the models we have, which very accurately describe the behavior of things like electrons etc, describe the true nature of particles. And of course, like all of our information of reality, we can’t know for certain what the true nature of anything is.