Often, when doing circuit analysis, any current that enters one of the capacitor's plates is assumed to exit the other plate.
We can assume this because when we inject an electron on one plate, the field it produces will repel other free charges around it. If the nearest free charges are on the other plate, then those are the ones that will get repelled, leading to the current out of one terminal being equal to the current in the other.
Of course you can also arrange, for example, for both plates to have some potential relative to your reference ground node. If a net charge moves in or out of the capacitor to change this potential, then you would model that with a parasitic capacitance between the two terminals of your capacitor and some other location in the circuit. This parasitic capacitance would account for electric field lines that go from the capacitor structure to "somewhere else" rather than originating on one plate and terminating on the other.
one of my motivations for studying this is for high-frequency circuits
In high frequency circuits you won't be assuming that a metal object is an equipotential. If you make your two "plates" larger than ~1/10 of the wavelength associated with the highest frequencies in your circuit, you will create a distributed structure rather than a lumped one. If the "plates" are very long and skinny, you have made a transmission line, for example. Then you will find that signals propagate along the structure as waves, with behavior dictated by the balance of the capacitance and inductance of the structure.
At some level you should also remember that all of our lumped circuit analysis is an approximation, based on certain simplifying assumptions about the nature of the circuit. If the lumped circuit model of a capacitor isn't adequate for explaining some particular circuit or device, you may have to perform a more detailed analysis, for example using Poisson's equation to analyze an electrostatic structure, or Maxwell's equations to analyze situations where magnetic and electric fields interact with the structure of the circuit (i.e. high-frequency situations).
There are two ways to look at this
One is that opposite charges attract each other. So if the bottom half (A) is charged relative to ground, and even more charged relative to the top left half (B), then whatever charges we find in A will be most attracted to B and will migrate as close as possible to them by electrostatic attraction.
We could imagine A is positively charged, B negatively, and the upper right, C, is uncharged. The charges in A are most attracted to B as C has no net charge, so they aren’t attracted to C.
The other way to look at this is that whenever charges are free to move, they “run downhill” to lower energy potential states. The charges will migrate to the configuration of lowest potential. The entire section A and entire section B will still be charged wherever the charge is, to they do comprise a capacitor. So where do the charges go to minimize $V$ across this A - B capacitor?
To minimize $V= \frac{q}{C}$ we maximize $C$, because $q$ is not changing. Because $C=\varepsilon_0 \frac{A}{d}$ for a plate capacitor, we can therefore deduce that maximizing $C$ means lowering the distance between charge (because for a plate maximizing $C$ would mean lowering $d$).
But we also know that potential is the work per unit charge to create this configuration. To minimize potential we minimize how much $\int F dx$ would be used to set-up the separated charges. Clearly they should be as close as possible for that. They’d all be on the left plates.
Best Answer
There's no reason the sides have to be equal, but if they aren't, the capacitor obviously has a net electric charge. Moreover, the electric field lines emanating from the capacitor have to go somewhere, such that the whole capacitor is also one half of a larger capacitor. In a circuit model, you would simply represent this as two or more separate capacitors, each individually balanced with zero net charge.
If the net charge of the entire circuit is nonzero, then you have to add a capacitor with a terminal going "nowhere," to a node representing the outside world. The capacitance of this depends on environmental conditions, i.e. the dielectric constant of air, and the shape of the electric field (particularly the surface area of the exposed metal).