Helmholtz Coil — What affects uniformity? Does a high current affects the uniformity of the $B$ field in a Helmholtz coil? For example, I was wondering if it would affect the width of the flat region of the net magnetic field as seen below.  If not, what factors could affect/increase the width of the flat region other than the distance between the two coils?

A second Question would be how would a non uniformity affect a circular beam of electrons within the field -  for example would the circle become non uniform?
 A: For the first question the intensity of the magnetic field would change but to shape of the magnetic field would stay the same assuming the current increased in each coil equally and the coils were identical etc.
The second answer depends on the details. Since the field is not perfectly uniform you would need to calculate it out.
A: The magnitude of the current has no effect on the (relative) uniformity of the field.  This is a consequence of the Biot-Savart Law,
$$\vec{B}(\vec{r})=\frac{\mu_{0}}{4\pi}\int_{C}\frac{I\,d\vec{\ell}\,'\times(\vec{r}-\vec{r}\,')}{|\vec{r}-\vec{r}\,'|^{3}}.$$
(The integration path $C$ follows the circuital path, in this case encompassing both loops.)  The key thing to observe about the expression for the magnetic field is that it depends on the current only through an overall factor of $I$; everything else is composed of fundamental constants and purely geometrical expressions.  Therefore, changing the current has no effect on the three-dimensional shape of the magnetic field; it just scales the magnitude of the field vector $\vec{B}(\vec{r})$, while leaving the ratio of field strengths at different points $|\vec{B}(\vec{r}_{1})|/|\vec{B}(\vec{r}_{2})|$ unchanged.
Whenever you have two idential, parallel, coaxial (normal to the $z$-axis) coils, each carrying the same current, there will be a region in the center of the configuration where the combined field from both coils satisfies $\partial\vec{B}/\partial z=0$—that is, where the field profile along the $z$-axis is flat.  How large the region in which the field is approximately constant is (both along the $z$-direction and in the $xy$-plane) depends on how far apart the rings are.  The specific "Helmholtz coils" configuration, with the separation between the coils equal to the coil radius $R$ maximizes the size of that region, by ensuring that not just $\partial\vec{B}/\partial z$ vanish at the center point, but higher derivatives as well; in fact, the first three derivatives of $\vec{B}$ with respect to the $z$-coordinate vanish there if the coil separation is $R$.  So with just two coils, the Helmholtz coil configuration is the best that you can do.  You can buy an off-the-shelf undergraduate-level experiment kit for testing the uniformity of the fields as a function of the coil separation.
If the level of uniformity offered by Helmholz coils with separation $R$ is not sufficient, it is generally worth using a full solenoid to generate a more uniform field.  In the limit of an infinite, tightly-wound solenoid, the field inside is completely uniform (not merely approximately uniform as for the Helmholtz coils).  For solenoids of finite length, the uniformity is obviously again approximate, but near the center of a long circular solenoid, a very, very high level of $\vec{B}$ uniformity can be achieved.
