Why does the nucleus not repel itself? If the nucleus is densely positively charged, why don’t the protons in the nucleus repel from each other and move towards the orbiting electrons? 
Because each proton is not only being repelled by the other protons, it is also being pulled by the oppositely charged electrons 
Why don’t these conditions make the atomic model impossible? I understand that electrons are in energy levels start so cannot force their way into the nucleus, but why not the reverse? 
 A: In navy nuke school, the analogy for the strong nuclear force was Velcro. I.e. attraction but at "touching" distances only.  
The strong nuclear force is sort of like a "glue" of neutrons that hold multiproton nuclei together.  There are certain ratios of neutron to proton amount than seem to work best.  See this chart:  https://en.wikipedia.org/wiki/File:Isotopes_and_half-life.svg  Note the rough 1:1 pattern but with a bit more "neutron glue" needed as you get to heavier elements.
There are also patterns of odd/even preferences and "magic numbers" (filled nuclear shells) which shows things are more complicated than just the "glue" idea, even if that holds as a first level insight.  See https://en.wikipedia.org/wiki/Even_and_odd_atomic_nuclei and  https://en.wikipedia.org/wiki/Magic_number_(physics) 
Electrons do enter the nucleus (have some wave function in there).  They can get captured in a decay process called electron capture, albeit rarely.  See http://wtamu.edu/~cbaird/sq/2013/08/08/why-dont-electrons-in-the-atom-enter-the-nucleus/
I would also point out that free neutrons are unstable:  https://en.wikipedia.org/wiki/Neutron#Free_neutron_decay  So, you probably don't need to worry about some sort of "electron capture death of the universe" where everything ends up neutrons.
A: There is another fundamental force of nature apart from the electromagnetic and the gravitational force. This is the strong nuclear force. Its presence is in between the interactions of protons and neutrons themselves or between protons and neutrons.
Unfortunately, the strong force has no macroscopic effect as to feel the interaction themselves because the typical range where they are stronger than the electromagnetic interactions is at the range of femtometres ($10^{-15}\ \mathrm m$). 
At such ranges, the strong force is stronger than the electromagnetic repulsion between the protons to hold them together. 
As for your second question on the nucleus itself travelling to the electron, if you think in terms of the centre of mass, the nucleus has higher mass than the electron and so the centre of mass of the system would be closer to the nucleus than it is to the electron. But in this case, the centre of mass is probably within the nucleus itself which is why it is a feasible idea to say that the electron reveolves around the nucleus. Although it is correct to say that the electron revolves around the combined centre of mass of the system.
EDIT: There is something that I have to add on for completeness -
As @dmckee has pointed out in his comment, the strong nuclear force is not fundamental by itself (I apologise) but is instead the result of a fundamental strong nuclear interaction originating between the gluons and the quarks that constitute these protons and neutrons. 
But essentially the strong nuclear force arises from this interaction. And hence the nucleus is stable from electromangetic repulsions due to the charge the protons carry.
A: The electrostatic repulsion force is long-distance, and the nuclear attraction is short-distance. So, protons do repel, and this is precisely what makes really large nuclei unstable. 
Secondly, electrons in the S-wave orbital have nonzero wavefunction at the nucleus, so effectively they are able to penetrate into the nucleus, and that is what makes reverse-beta decay possible.
