Do electrons in the same orbital but different spin feel each other's Coulomb repulsion? Or is it irrelevant, as orbitals are QM while Coulomb interactions are classical physics?
What I think I understood of orbitals is that particles with the same quantum numbers cannot occupy the same space (pauli exclusion principle), while with different numbers such as spin they can. Does that mean the two electrons are invisible to each other or just that they form a third entity that occupies an orbital?
By third entity, I mean a collaboration or synchronization between the two electrons.
I found this document, but I am unsure of the conclusion to draw: http://magnetism.eu/esm/2013/slides/lacroix-slides.pdf
 A: Yes, there is Coulomb interaction, which also leads to correlation in position.
As an example, you could look at helium. The binding energy of one electron is 4 Rydberg = 54.4 eV. But the ionization energy of neutral helium is 24.6 eV. 
Calculating this number is not so easy because it is a three-body problem. One way of taking into account  electron-electron correlation is by "configuration interaction" with higher orbitals. Or one can use density-functional theory.
A: Loosely speaking, yes. By 'feel each other's Coulomb potential' you mean that the behaviour of one electron is influenced by the presence of the other owing to electrodynamic effects. That is undoubtedly the case. If you were to model the behaviour of an electron in an atom by considering only the potential due to the nucleus and electrons in other orbitals you would calculate imprecise answers. To help you visualise the reasons, imagine two atoms, identical except that one is ionised by missing an electron that would normally fill an orbital; clearly the two atoms appear different to an electron passing by, and the difference must be attributed to charge effects.
To try to illustrate the point from another direction. Suppose that an electron in a given orbit did not 'feel' the Coulomb repulsion from the other electrons in that orbit. In that case the electron would feel only the Coulomb potential from the nucleus. If that were true, then a nucleus would attract an infinite number of electrons.
Another illustration is provided by the Hartree-Fock method for calculating energy levels, in atoms for example. In that method, the Schrodinger equation is solved for a single electron by considering a Hamiltonian that models the presence of, including the Coulomb interaction from, the other electrons orbiting the nucleus. 
In truth the Pauli exclusion principle is a post-hoc rule to reflect the observed population of electron orbitals.
A: 
...particles with the same quantum numbers cannot occupy the same space (pauli exclusion principle), while with different numbers such as spin they can. Does that mean the two electrons are invisible to each other<...>?

No, this doesn't mean that they are invisible. It's just that Coulomb potential is a "soft" potential: due to Heisenberg uncertainty principle the electrons have nonzero probability density at the point of collision, despite having infinite potential energy at that point.
Had the potential had a higher power in the denominator, e.g. $r^2$ instead of $r$, like in centrifugal effective potential, the electrons would then never be able to come arbitrarily close to each other, regardless of their spins.
See also my answer to the question "Can two electrons never touch each other?".
