Then does all charges, no matter how much charged are they, ionize the material around them since distance can get smaller and therefore electric field gets bigger. Consequently, the electric field exerted by the charge would be higher than the electrical breakdown limit of the material around it.
Let's say you magically stick a (positive) charge inside a piece of material. You can imagine getting as close to a charge as you want, but the "material around it" is still made up of atoms which are a certain distance away. The atoms of the material are not a continuum which gets infinitesimally close to your charge. The electrons of the material may get close, but you have to think of the electrons quantum mechanically: they are "smeared out" in space, and to avoid getting into that here, the small region where the electric field is arbitrarily high gets averaged out from the point of view of the smeared electron.
2-My other question is more basic. When we try to calculate the electric field between a capacitor which has two oppositely charged object, object charged with negative sign would exert a negative electric field whereas object charged with positive sign would eert positive electric field. If the amount of charges were the same in each object, would the positive electric field and negative electric field cancels out each other and would the net electric field be 0? I know it doesn't work like that but I want to get the correct information.
You have to consider not just the magnitude, but also the direction of the electric field. Say the positive charge is above and the negative charge is below. The positive charge creates an E-field pointing away from itself, so, inside the capacitor, that points downward. The negative charge creates an E-field pointing towards itself, so, inside the capacitor, that points downward. Thus both E-fields actually add together to make a stronger one. If you play the same game but outside the capacitor, you'll find the E-fields cancel each other (all of this assuming you do the standard infinite capacitor plate).
Electrostatic refers to the case where the fields are not time dependent. In that case the Maxwell's equations reduce to:
$$\nabla \cdot E =\frac{\rho}{\epsilon_o} \\
\nabla \times E = 0 \implies E=-\nabla \phi \\
\text{then,} \nabla \cdot \nabla \phi = \nabla^2 \phi = -\frac{\rho}{\epsilon_o}
$$
The solution to the last equation is:
$$
\phi = \frac{1}{4\pi\epsilon_o} \int \frac{\rho(\mathbf{r'})}{|\mathbf{r}-\mathbf{r}'|}d^3r'
$$
which gives the electrostatic electric field equation you have written for point charges.
But if the case is time dependent then you have the Maxwell equation:
$$\nabla \times E = -\frac{\partial B}{\partial t}$$
and you can no longer define $\phi$ as we did above. In this case you have to work with retarded potentials:
$$\psi(\mathbf{r}, t) = \frac{1}{4\pi\epsilon_o} \int \frac{\rho(\mathbf{r'}, t')}{|\mathbf{r}-\mathbf{r}'|}d^3r'$$
where $\psi = \phi, A_x,A_y,A_z$ and $t'=\frac{|\mathbf{r}-\mathbf{r'}|}{c}$. (Note that when $\psi = \phi$ then $\rho$ is the charge density and when $\psi = A_i$ then $\rho = J_i$, i.e., the $i^{th}$ component of current density). Then the fields are given as:
$$
\mathbf{ E} = -\nabla \phi -\frac{\partial \mathbf{A}}{\partial t}\\
\mathbf{B} = \nabla \times \mathbf{A}
$$
These time dependent fields turn out to be very complicated expressions and are named the Jefimenko equations.
Best Answer
Think of a field as "the effect something has which spreads out". Like rings in water spreading from the splashing stone or like sound spreading from the clapping hands.
A magnetic field is the effect of a magnet. The field spreads out from/to the poles and grabs everything on its way that will interact magnetically.
Gravity is the effect of things with mass. The gravitational field spreads out from the mass and grabs every other mass on its way.
And an electric field is an effect of a charge. The field spreads out from/to charges and grabs every other charge on its way.
Everytime I said "grab", what I mean is that this field converts into a force of some sort. A force that causes motion in those other interacting objects.
A field is a way to explain how forces can be applied without physical touch. A charge is pulling in something else with an electric force without touching it - the "stuff" that connects the charge with its force through empty space is the electric field.