If I am not wrong, the Michio says that an electron (not the parts of an electron) can be found in many places at the same time. Is that right?

If that is the case, consider a gaussian surface enclosing the electron at many places at the same time, then the charge inside the surface will be integral multiple of e (i.e ne, where n>1). But, we know that charge on electron is just e.

This doesn't seem to allow electron to exist in many places at the same time. So, can an electron really exist in many places at the same time?

I think your yes / no question is a false dichotomy and you are unlikely to get many people to commit themselves either way. Fact is that the electron can sometimes be accurately located and sometimes it can't. You only know it's there if you detect it but you cannot say where it is without looking. You can chose or not whether that means it's in several places at once - before you find it. We are in the same neck of the woods as Schroedinger's cat here.
Your gaussian surface is a classical concept so I don't think you can validly come to the conclusion you have done.

Kaku is better at science than at explaining science. We spend a fair amount of time here unconfusing people who have been confused by his oversimplified explanations.

There are a bunch of threads over in the quantum mechanics forum about what that electron "cloud" does and does not mean. Roughly speaking, it does not mean that the electron is in many places at once, it means that the electron is nowhere until we precisely measure its position - and obviously you cannot draw a Gaussian surface around it until you have localized it to inside the volume enclosed by that surface. But once you have localized it to that volume, even though it still doesn't have a position more definite than "100% of the cloud is confined within the volume" we do know that there's exactly one electron's worth of charge inside that volume.

Are you familiar with the Born interpretation of the wavefunction? The wavefunction does not say that the electron is everywhere at once. It gives you the probability of finding the electron at any one location in a quantum measurement. If you find it in one place, then you won't find it in any other place using the same measurement. (Future measurements, of course, have an new probability of finding the electron in various places.)

If you apply the Born interpretation to a Gaussian surface, you can calculate the probability of measuring the electron inside that surface, so the total enclosed charge is either 0 or 1 (units of -e).