Is gravity non-negligible compared to the electromagnetic force?

Gravitation is a negligible effect unless you have a large number of particles and a very small percentage of net charge.

For two protons, the electromagnetic force is $10^{36}$ times stronger than their gravitational force. But if you have a small object made up of protons and neutrons and electrons, with a total of about $10^{36}$ atoms and just one more proton than electrons, then the forces are equal. For planetary, stellar, and galactic sizes the electromagnetic force can be ignored compared to gravity, except when you have localized overcharges like around solar flares.

For atoms and molecular effects, or even electrical circuits, gravitation can just be ignored

See relative strengths of the elementary 4 forces at Wikipedia page on fundamental interaction.

For two individual electrons, the electric force will be much more powerful at all points.

For example, suppose they are 1 meter apart, the gravitational force on each electron will be (the fact that they are moving does not change the force):

$$F_G = \frac{Gm_e^2}{r^2} = \frac{6.674\times 10^{-11}\times 8.297\times 10^{-61}}{1^2} = 5.538\times 10^{-71} \:\mathrm N$$

The electric force will be:

$$F_E = \frac{kq_e^2}{r^2} = \frac{8.988 \times 10 ^ 9 \times 2.566 \times 10 ^ {-38}}{1^2} = 2.306 \times 10 ^ {-28}\:\mathrm N$$

We can see that the electric force is several magnitudes higher than the gravitational force.

Although eletromagnetic force tends to be higher in magnitude then gravitational forces, the presence of negative and positive charges neutralizes the effect of this force... Gravitaional force on other hand, is additive and increases propotionaly in magnitude with mass.