He found that his equations worked perfectly for the everyday negatively charged electrons, but they also worked for electrons with the opposite electric charge.

The actual problem at the time of Dirac was that the energy of the positron with negative. This was one of the issues of the Klein-Gordon equation that Dirac wanted to solve. And he didn't mange. The states with a negative energy were there. We had to wait a little bit before being those identified with the positron.

They are not actually tiny little ball (as we sometimes imagine), but quantum mechanical objects.

Actually, the reality is worse: they are quantum field theory objects. :p

The total electric charge of a particle before and after collision has to be the same. Why is that necessary? We don't actually know. We observed this pattern in real life experiments and incorporated these rules in our theories.

I tend to disagree with your statement. We have a gauge symmetry that implies electric charge conservation. Why we have the gauge symmetry in the first place is actually the real question. However, as soon as we impose the symmetry, the conservation law is automatic thanks to our good friend Nœther.

There is yet another law in our universe about smooshing, that says that "electronness" and "muonness" (basically quantum properties) has to be conserved.

This is not totally correct. Those quantum numbers are known to be violated in very-high energy processes. Those symmetries are actually called accidental symmetries. We didn't impose them at the first place. They kind of come for free, and there is thus no good reason for the symmetries to be maintained in every energy regime.

To finish, a funny comment: do you know that the LHC picture you used is the one of CERN April fool's article of two years ago, stating that CERN has discovered the force. The Force, like in Star Wars :D (in short, another image may have been a better choice :p )