Is it possible to have a particle with a positive and negative charge? When you have two oppositely charged particles, according to Coulomb's Law $F=k q_1q_2 /d^2 ,$ the closer they are together, the greater the force they exert on each other. On some simulations I´ve seen, if the particles are lined up directly on each other, the force excerpted on each other is infinity. This doesn't work with Coulomb's Law because the distance between the center of each particle would be zero. If the scenario where the two particles of opposite charge were in the same position, would that be considered one particle with two different charges? Or would it be similar to quantum computing where the bit can be 0 and 1?
 A: To understand particles at the level of very small distances requires at least a little quantum mechanics.
Consider positronium. This is a bound state of an electron (charge -1) and a positron (charge +1). It has a very short lifetime, 0.1244 ns. The two particles annihilate and release photons. This can happen because the quantum numbers of the two particles are exactly opposite. They are a particle-anti-particle pair. So all the conservation laws can be maintained if they convert to photons.
Compare to a hydrogen atom. This is a bound state of an electron (charge -1) and a proton (charge +1). But in this case, the proton is not the anti-particle to the electron. Not all quantum numbers are opposite. So they cannot annihilate, the system is stable.
But what about the infinities when the electron charge and proton charge overlap? Quantum mechanics tells us that a particle does not truly have zero size, even when it is a point particle. "Point particle" means it has no sub-structure. So far as we can tell, an electron has no component parts. But a proton is made up of three sub-components called quarks, so is not a point particle. Just as the hydrogren atom is not a point particle because it is made up of an electron and a proton.
Because particles are not zero-sized, they avoid the infinity. This particular one anyway. Instead of the potential having this infinitely deep well at the origin, it has a little ball of distributed charge. So for the very small fraction of time the electron spends overlapping the proton, it sees a finite constant value of potential.
There is a very great deal more to the situation than that. If you are interested, you should dig in to some quantum mechanics texts. If you want to go that way you should be paying attention in math class as well. There is a very large amount of calculus awaiting you in QM.
A: One has to define the term "particle". After more than a century of observations and measurements, mainstream physics uses theoretical models to answer such questions, but these models depend on the frameworks used to describe the particles.
There is the classical framework which you use in your question, but that framework is limited to dimensions larger than nanometers and energies measured in joules. When dimensions get very small, as in your hypothesis of overlapping two particles, one is in the quantum mechanical frame and has to use the quantum mechanical theories to model the data and predict what happens.
In particle physics at the quantum level, the elementary particles are point particles, and their (x,y,z) follows quantum mechanical probabilistic equations and an overlap, for example of an electron with a positron, leads to annihilation, disappearance of charge,  and just  gamma rays , if the energies are low.
All other particles are composites of elementary particles  again governed by quantum mechanical probabilistic equations, and as quantum composites it has no meaning to talk of the charge being in a specific  (x,y,z). When a proton meets an antiproton  , complicated quark interactions happen, depending on the energies ,  and charge disappears.
So the classical infinity of Coulomb's law is never reached at the underlying quantum level.
A: If the particle is composite, then yes. For example a dipole.
If the particle is elementary, then no. However, if the charges are of different kinds then it is possible. For example, the hypothetical dyon has both magnetic and electric charges.
A: Other's have mentioned important quantum mechanical points about the matter of opposite charges approaching each other. There's a classical matter that helps approximate electric dipoles.
Suppose you have a charge q at $(0,0,z_0/2)$ and -q at$(0,0,-z_0/2)$. It has a dipole moment of $qz_0\hat{k}=p_0\hat{k}=\vec{p}.$
If you take the limit of the resulting electric field as $z_0\to 0$ while $p_0$ remains constant (by adjusting the charge appropriately), you get the field of an ideal electric dipole:

Notice, there is a field even though there is no net charge. This is a decent approximation for the electric field of polar molecules. So classically, you could say opposite charges near each other give rise to a dipole moment with a characteristic Electric Field following an inverse cube law, as opposed to the inverse square law of a standalone electric charge. The charge is zero, but dipole moment, quadrapole moment, and so on are not necessarily $0$ and give rise to Electric Fields.
A: Electric charge is a property of matter. Matter has mass and occupies volume. It is not possible for different matter to occupy the same space at the same time. Two particles of opposite charge can get very close to one another, but they can never occupy the same position simultaneously. Simply put, "the scenario where the two particles of opposite charge were in the same position" cannot occur. Trying to reason about such a situation would require matter with zero volume, defying the fundamental properties of what matter is.
