I plan to open the dataset once it's collected and produce material for school pupils to support their science and statistics classes. It only takes a few seconds to join in, so please help out!
What we do know is that it, electromagnetically, has a magnetic dipole, a charge monopole, and - as best we can tell - no charge dipole. The electromagnetic properties are the shape in the only sense that anyone ever really uses the word shape. So its shape is round.
If you want a picture in your minds eye of what we know about the electron, with the understanding that this is just a picture and the theory is the authoritative description, then a ball of charge is, in my opinion, closer to our current theories than a mysterious point particle. You bring up the classical charge radius above: this is precisely the radius that electron-positron virtual pairs can be created, and in a meaningful sense is something like an electron radius. (Although note that they have set limits on the charge radius by comparing the measured magnetic g-factor with the QED calculation, this is the 10^-20something that wikipedia quotes, but this really measures something different - I think you need to be more specific than just saying size at this length scale).
I understand the complications due to the electrons being in molecules - if you haven't twigged yet, JonyEpsilon == J.J. Hudson == first author of paper under discussion - but it doesn't have any real bearing on what we're talking about.
Hope that makes sense
If you look at history, you can see that there is essentially a blueprint for turning an open society into a dictatorship. That blueprint has been used again and again in more and less bloody, more and less terrifying ways. But it is always effective. It is very difficult and arduous to create and sustain a democracy - but history shows that closing one down is much simpler. You simply have to be willing to take the 10 steps.
To understand a program you must become both the machine and the program.