Considering this hypothesis...is charge really quantized? [If anything goes against any mathematical or physical rules please let me know. I am a first year undergraduate student perusing a joint major in mathematics and physics so I do not have a complete background in those fields. I am just using my imagination and what I already know]
Consider an isolated electron in space. We know at all points in this space an electric field exists due to this electron (except perhaps at infinity).
Now, from Coulomb's Law we can find a vector for the electric field  due to this electron at all points in this space.
Since this is possible, does this imply the charge of the electron constitutes an infinite number of very small charges 'dq' that each produce a linear field where, each oppositely positioned dq destroys the field within the electron (shown in attached picture)?

 A: The electron is an elementary particle in the underlying building blocks of matter organized in the elementary particles table of the standard model of particle physics. Elementary particles are point particles.
The standard model is a precis of a very large number of measurements (data)  fitted by mathematical models of theoretical physics. A point particle has no charge distribution, its charge is given in the table in the link.
To understand elementary particle physics one needs quantum mechanics and this cannot be done except through serious study in the appropriate courses.
A: Now, from Coulomb's Law we can find a vector for the electric field due to this electron at all points in this space.
When you read about Coulomb's law, you can see that it describes the force  between two charged particles. When you set them down such that they have no initial relative motion, they move together or apart in a linear fashion. But note that the radial arrowheads in your picture don't really "work". Two electrons move apart, two positrons move apart, and an electron and a positron move together. Ergo those radial arrowheads don't depict force. Moreover they don't depict field, because the electron doesn't have an electric field, it has an electromagnetic field. The force between the electron and the positron is the result of two electromagnetic fields interacting. See section 11.10 of Jackson's Classical Electrodynamics where he says "one should properly speak of the electromagnetic field $F_{\mu\nu}$ rather than E or B separately". IMHO it's a pity this isn't in section 1, but such is life.   
Since this is possible, does this imply the charge of the electron constitutes an infinite number of very small charges 'dq' that each produce a linear field where, each oppositely positioned dq destroys the field within the electron (shown in attached picture)?
No. You're barking up the wrong tree I'm afraid. There's a linear force when the two electromagnetic fields interact, but there is no linear field. I know that's what you can find in physics courses, but it's misleading. The field is the electromagnetic field. Ask your course tutor to depict it for you. Tell him about this picture by Maxwell on page 7 of this paper:
 
You could also mention the gravitomagnetic field I suppose. But anyway, once you have a concept of the electromagnetic field, I'm confident you won't feel tempted to hypothesize about a charged particle's charge being made up of small charges.
