> What about the particles that randomly pop in and out of existence?

I like to imagine they're somehow just an observational error, otherwise the https://en.wikipedia.org/wiki/One-electron_universe is real and we get a universe-sized '—All You Zombies—'

> How can two electrons actually repel each other

Indeed. I think it's something we can only intuit, I don't think we've really gotten to the bottom of it. Trying to push two electrons together feels like trying to push a car up a hill, or pressing on springs. The force you fight against is just there and you feel its resistance