Yes, it's weird. Quarks and electrons are just points, they have no interior, so they can't absorb other thing. I'm not sure about the official explanation, but IIRC they old quark and the electron just disapears, and a new quark appear instead???

Actually, it's more complicated, because there appears also a new neutrino, it's something like: quark up + electron --> quark down + neutrino. But neutrinos are very difficult to see, so let's ignore it.

Actually, it's more complicated, because there is an intermediate W+ or W- particle. The W particles live for a very short time, so you can ignore it unless you work in a particle ascelerator, but to get the correct results you must use the W+ or W- particle in the middel of the transformation. The correct equation are

quark up + electron --> quark up + neutrino + W- --> quark down + neutrino

quark up + electron --> quark down + electron + W+ --> quark down + neutrino

Actually, it's more complicated, because sometimes you get a strange quark instead of a down quark. Ignoring the W+ or W- particles

quark up + electron --> quark strange + neutrino

(And perhaps there are a few more "actualies", but these are the most important weird cases.)

> How exactly does that happen?

Magic? Nobody knows, we only can make experiments and guess the equations. There is people that enjoy discusing the interpretation, but I think it's better to ignore it and enjoy the agreement of equations and experiments.

> Doesn't each particle species have it's own separate quantum field?

Yes

> How does one convert into another?

Magic? Sorry, not good answer.

> Electromagnetic field can convert into quantum electron field by spawning electron-positron pair from a single photon. But all those exchanges are just weird. It's really shocking that people managed to figure out the math that rules over this.

A lot of the rules can be understood using magic balls. You can learn the rules to combine particles and draw Feynman diagrams as if they were magic balls. It took a long time to discover them, including two or three Nobel prices. It was very difficult, but somehow it was possible to discover one rule at a time, and add another rule a few years later, that made the whole procces slightly easier.

The equations are more difficult. I only learned a small part of them (and I had forgoten them). Also, they were discovered in small steps, sometimes as simplification or generalizations of previous equations, and had a few Nobels in the list of discoverers. Also, most particles have the same equations, for example all the quarks have the same equations so you just copy it 6 times, but photons have a different equations. And you have one big equation that combines all of them in a very long formula, but each part is simple.