How exactly do protons and electrons interact with each other? How do these particles exchange information about charge and position between themselves, even though there's mostly empty space between them?
Also what happens if a free electron passes closer to a nucleus of a hydrogen atom than its own electron? 
My random guesses so far:


*

*Proton won't react because it's somehow locked with its existing electron

*Proton will catch it and release the previous one 

*Proton will catch it and keep both of them with half the force

*Existing electron will simply repel incoming one

*Something else?

 A: The proton and electron exchange information via a gauge boson, in this case, a virtual photon. This is how the electromagnetic interaction is mediated.
As for your other question, the electron will get decelerated and deflected and emit a photon, releasing some of its energy in a process called Bremsstrahlung
A: C: See "Quantum Mechanics". More specifically you're wanting solutions to the Schrodinger Equations that represent your system. In this case that is an electron represented by a probability density function $$  \psi (\vec{x},t ) $$ under a potential $$ V(\vec{x}) $$ which is the potential energy function of the system. This contains information regarding the relative distances and charges of the particles in the system.
Once we find solution  $ \psi(\vec{x},t), $ for both space and time variables for $$ i\hbar\frac{\partial }{\partial t}\psi=\hat{H}\psi $$ where $ \hat{H} $ contains information regarding the energy of the system from the aforementioned potential $ V(\vec{x}) $, we have a time evolutionary model of the system! 
This allows us to create a 3 Dimensional moving picture of the probability density of the electron in this system. In other words, we can see the various locations this electron can be with respect to the hydrogen atom. The plot would not be of $ \psi(\vec{x},t) $, but of $ |\psi(\vec{x},t)|^{2} $ per the rules of the underlying mathematics. Knowing the energy equations of motion we can calculate the Euler-Lagrange Equations of motion for the system. These turn out to be the equations of motion that would answer your questions as to how your system would interact. 
From there we can calculate the corresponding momentum equations of the system and arrive at the same functions for motion aforementioned. For a more rigorous introduction to the course I recommend Mathematics for Quantum Mechanics: An Introductory Survey of Operators, Eigenvalues, and Linear Vector Spaces (Dover Books on Mathematics) by John David Jackson or Lectures on Quantum Mechanics by Paul Dirac.
For a more conceptual introduction I recommend The Quantum World: Quantum Physics for Everyone by Kenneth W. Ford and Diane Goldstein or The Quantum Universe: (And Why Anything That Can Happen, Does) by Brian Cox.
Happy reading, scholar!
