It is incorrect to imagine the electron having a shape, and it is incorrect to imagine electrons orbiting the nucleus. Scroll down to the "orbitals table" to see what an atom with a single electron looks like, for various values of energy and angular momentum:

In relativistic quantum mechanics we find it necessary to make the electron a point particle, since otherwise it would tear itself apart with internal forces. But this is not something that anyone can or will ever measure, it is just part of the mathematical framework.

This was already addressed, but electrons don't have a shape, rather they're referring to the shape of their pattern of motion, or rather, density. The shape of the space that they occupy is what's called an orbital.

It's the other way around, really. First, electrons repel each other since they have the same electric charge (negative). Second, there's a 'rule' in quantum physics (the Pauli priniciple), that dictates that they can't occupy the same point in space (unless they have opposite 'spin).

So, the shape of the orbitals are largely dictated by these effects.