If you really treated them as points the theory would blow up because the electrostatic potential energy of a point charge is infinite. This is dealt with by “renormalization” which is roughly: assume the theory isn’t really valid all the way to a zero length scale and that we can average out everything that happens below some cutoff size and that it doesn’t really matter where we place the cutoff because the theory works the same if you change the cutoff and change the other parameters accordingly.

They are, but there are some interesting theories about their internal structure. Be aware that this is probably junk science but I have currently been enjoying this effort to describe them via the em field.

https://quicycle.com/understanding-electrons/

And the video essay on the subject https://www.youtube.com/watch?v=hYyrgDEJLOA (Huygens Optics: Williamson & Van der Mark electron model | Are electrons made of light?)

Points are a realist view - there's a real object there.

Although some physicists disagree, QM slants very anti-realist. There are no objects anywhere, no particles, no waves, only probabilistic interactions, some of which can be snapshotted into localised partially definite results.

So there are only interactions between probability distributions in space and time, and "particle-like events."

No pointy objects, and no need for them.

Kind of, but not really.. though there are simple models with electrons as a point charge, a more accurate model involves the electron field describing the probability of an electron existing at any region in space (not to be confused with the electromagnetic field, the medium in which photons propagate).