As far as I have observed, all the expressions for fundamental forces (at least in Newtonian Physics and E&M) have no explicit dependence on time. Why is this?
For eg: Electric force depends on charge and electric field, magnetic only on velocity, gravity on distance.
We can't say why nature behaves as it does. We can only describe how it behaves. But we clarifying what time independence means will help.
If you set up two charges a certain distance apart, you measure a certain force. If you come back later and set up the same situation, you expect for measure the same force. The laws of physics have not changed. This implies that the nature of charges and the nature of space have not changed.
Over cosmological times and distances, we do see changes. Space is expanding. That doesn't mean things are moving apart. It means the nature of space is changing. The metric is changing. It is difficult to describe the difference between the two. But there are measurable consequences, such as the red shift of light from extremely distant objects. See If spacetime itself is expanding, how could we ever tell? or Can space expand with unlimited speed?
One deeper reason behind this can be the time-translation symmetry. The law :
$$m\frac{d^2 \vec{x}}{dt^2}=-\nabla \frac{kq_1q_2}{|\vec{r_1}-\vec{r_2}|}$$
has a few symmetries built into it:
The acceleration depends on ($\vec{r_1}-\vec{r_2}$), i.e. the relative position vector. This is translational symmetry. If you displace the entire system by the same amount, the relative position vector remains unchanged, and hence the force remains unchanged. Hence, the laws of physics treat all locations in space equally.
The magnitude of acceleration does not depend on the direction of the relative position, but only on its modulus. This means you are allowed to rotate the system and the acceleration vector will also get rotated identically. This is rotational symmetry.
The acceleration is independent of time. The same experiment performed tomorrow will behave the same as if it were performed today. This is time-translational symmetry.
The forces do depend on space, but in a way to preserve space translational symmetry. Similarly, there is no time dependence to preserve time-translational symmetry
All these symmetries are broken in General Relativity, because spacetime becomes a dynamical entity and hence different spacetime points can have different properties (e.g. curvature) .
There is no formal reason for that. In fact you could imagine a theory in which fundamental constants do depend on time. You can consider Dirac's large numbers hypothesis (https://en.wikipedia.org/wiki/Dirac_large_numbers_hypothesis) as an example of time-dependent gravity.
In my understanding the fact that the world around us looks (more or less) exactly the way it used to a week ago is just a happy coincidence with no fundamental reasons enforcing it. Without this empirical fact, we, most probably, would never be able to develop a scientific understanding of the Universe.
You're not being consistent in your terminology, and that's causing confusion.
For example, you say that "Electric force depends on charge and electric field." The equivalent statement for gravity would be "Gravitational force depends on mass and gravitational field." (Newtonian) gravitational force doesn't depend on space (or time) any differently from the way (non-relativistic) electric force does. The "field" is just a way of describing the forces acting on local test charges.
You can replace electric charge with gravitational mass, and electric field with gravitational field, and you get closely analogous situations, at least as far as most undergrad physics problems are concerned. (Differences arise on close inspection for several reasons, most prominently because electric charge can have either sign, but gravitational mass always has the same sign. Gravity's also deeply interwoven with the nature of spacetime and is a spin-2 field as opposed to electromagnetism's spin-1 field.... but delving into those differences will take us pretty far afield.)
Also, you can generate magnetic fields with moving electric charges and such effects are also responsible for the "magnetic force"'s velocity dependence. The "magnetic force" is just an "electric force" with relativistic effects. (If you hear folks talk about the Lorentz force or electromagnetic waves, it's because they're conscious of the coupling between electric and magnetic effects.) You can even get a gravitational counterpart in systems with large masses moving at relativistic speeds.
These fields and forces fall off due to distance, but because of the finite speed (c) of electromagnetic and gravitational waves, there are also time delays so that they act not on the current displacement between charges, but on the displacement the charges had at a previous time... it's just that as long as everything's moving much slower than light, that slight delay doesn't make a measurable difference.
In summary: the forces do depend on the relative time between charges, but the time scale is very short, of order D/c, where D is the separation and c is the speed of light, with higher-order effects that scale with v/c, where v is the relative speed of the charges. If you track those time delays and higher-order effects explicitly, electromagnetic and gravitational effects seem more similar.
