What happens when we bring an electron and a proton together? I have a couple of conceptual questions that I have always been asking myself.
Suppose we have an electron and a proton at very large distance apart, with nothing in their way. They would feel each the other particle's field - however weak - and start accelerating towards each other.
Now:
1) Do they collide and bounce off? (conserving momentum) 
2) Does the electron get through the proton, i.e. between its quarks?
3) Do both charges give off Brehmsstrahlung radiation while moving towards each other?
Different scenario:
Suppose I can control the two particles, and I bring them very close to each other (but they are not moving so quickly as before, so they have almost no momentum).
Then I let them go:
1) Would an atom be spontaneously formed? 
2) If anything else happens: what kind of assumptions do we make before solving the TISE for an Hydrogen atom? Does the fact that the electron is bound enter in it? 
This is to say: is quantum mechanics (thus solving the Schrödinger equation) the answer to all my questions here?
 A: Part 1: Conceptual/physical intuition
Since there is an electrostatic attraction between the 2 particles, then when
they are apart they are at a higher potential energy then when they are
together. 
Here's an analogy:
Physically, this situation is like having a ball at the top of a hill
overlooking a valley or well. The ball will roll down the hill and that
potential energy is converted into kinetic energy. When the ball reaches the
bottom of the valley it will start climbing back out of the well and turn that
kinetic energy back into potential, so if the ball starts at rest it only gets
back to being as high as it started. However in the real world there is
friction that will steal some of this kinetic energy and so the ball will roll
back and forth, but eventually come to rest at the bottom of the hill.
For the electron an proton you'll see something similar. The 2 particles will
accelerate towards each other, pass/scatter off each other (and then repeat) and will slowly lose
energy to "friction" i.e. to radiation.
Part 2: Specific questions

1) Do they collide and bounce off? (conserving momentum)
  2) Does the electron get through the proton, i.e. between its quarks?

The collision between the two particles is perfectly elastic. In addition the
energies (~13eV) are so small relative to the strong force holding together the
proton that quarks are not involved in any way, and the scattering is described
by Rutherford
scattering.

3) Do both charges give off Brehmsstrahlung radiation while moving towards each other?

The 2 particles will radiate and lose their kinetic energies. The term
Brehmsstrahlung is generally reserved for much higher particle energies (>keV), and much larger accelerations. 

Suppose I can control the two particles, and I bring them very close to each
  other (but they are not moving so quickly as before, so they have almost no
  momentum). Then I let them go:
1) Would an atom be spontaneously formed? 

You can immediately describe the 2 particles by their center of mass
description (an atom) plus their individual attributes (i.e. what the particles
are doing within the atom). Assuming the 2 particles start off at rest, then
they are in a bound state already because they can't escape each other (go off
to infinite separation) due to lack of energy.
However the atom will not be in it's ground state until it has decayed into the
lowest level via spontaneous emission of radiation.

2) If anything else happens: what kind of assumptions do we make before
    solving the TISE for an Hydrogen atom? Does the fact that the electron is
    bound enter in it? This is to say: is quantum mechanics (thus solving the
    Schrödinger equation) the answer to all my questions here?

The TISE of the atom itself will give you energy levels etc, but you will not
get spontaneous emission into the ground state unless you put it in by hand
(and it would not be time independent anymore) or also quantize the EM vacuum
(which is how you derive SE). So trying to solve it would be like solving the ball moving on the hill while ignoring friction, it will just oscillate at constant energy forever.
A: Check this link here. This is a quote from the website:

When an electron falls from infinity towards a proton it acquires 13.6 electron Volts of energy to reach the ground state “orbital” around the proton. I have always wondered why it does not go all the way. Apparently, its Debroglie wavelength has to fit” around the “orbit radius” for it to occupy a stable state.
  Perhaps another explanation is that an electron can only arrive in an atom and occupy an orbital by dissipating its arrival energy in the form of a photon.

I believe they would emit Bremsstrahlung as there is acceleration involved, but the electron will still end up gaining 13.6 eV of energy anyway.
In the second case, the electron is inside a potential well, and will stay confined, thus creating an atom. Solving the TISE is very useful in this case.
A: To provide a graphic version of Punk_Physicist's answer, we have the Feynman diagram for that particular interaction:

This diagram evolves in time from bottom to top, ie take a piece of paper, and run it upwards along the diagram to see how the system evolves. We have an electron and a proton coming towards each other, then we see them interact by releasing a $ \gamma $ particle, ie a photon--electromagnetic braking radiation (See Bremsstrahlung). The energy of this photon is of course the kinetic energy lost by the electron passing by the proton. The closer they pass, the higher the potential, the greater the loss in $ E_k $ of the electron.
A: To rephrase: an electron approaches an cation.
Most likely result: a photon is emitted and the electron is captured.
Added: The way I see it: an ion and an electron at a large distance apart. This would probably be an electron in a very high orbital of the ion. As the electron approaches, all the natural forces affect the electron and the electron slows down when it approaches the nucleus. This would equate to dropping down orbitals which would release a photon.
A: In nuclear fussion electrons and protons can fuse to form neutrons with the release of photons.
