Are my intuitions about how batteries work right? Being really fascinated by how electricity actually works, I started studying batteries.
I initilally understimated the complexity of this apparently-simple objects, and thorugh studies and researches I have understood how they are the result of complex chemical and physical discoveries, blendend together in a triumph of engineering.
Anyway there are  some things that I still have to figure out completely, I made some pictures that aim to clarify my doubts and the question objects.
This is the picture of a zinc-copper cell with a non-specificated electrolyte:

It represents essentially how I imagine a cell of this kind.
Zinc and Copper have both reached a dynamic equilibirum with the electrolyte, so charges leave electrodes at the same rate as charges enter.
This creates an electrode's electric field, and the elctrode potential at the same time.
The net electric field represents the net force driving the elctrons from the zinc to the copper, resulting from the superoposition of the elctrodes fields.
Given that this analysis is right ( If it's not, please help me understand where is flawed), I don't understand how this electric field ( and so the potential) changes as time passes.
If electrons are leaving zinc to reach copper (and eventually react with some positive ions in the electrolyte,right?), how the electric field is mantained so that the flow can go on?
As electrons leave, how the zinc electrode field is affected?
Are positive ions leaving the zinc at a faster rate in order to compensate for th electrons deficency? (That would explain the zinc corrosion)
Is there an electrode's size limit (corrosion limit) where the electrons  that make up the elctrode aren't enough to keep the same voltage(the chemistry's votlage)?
Is it at the point that we start to see a decrease in the voltage of the cell?
Of course, all the considerations presume the precence of a salt bridge capable of mantaing the elctrolyte charge stable.
I'll assume you are someone less than 16 years of age, since usually it's around 16-17 years (11th or 12th grade) that you'll probably learn how batteries work.

You're assumptions about how electrons are taken from the zinc cell and moved to the copper cell are correct. But considering your setup, as the charge moves, an electric field and a potential difference would build up, won't it? In such case a battery would eventually stop functioning.

While batteries do stop functioning after a while, it's due to the reaction being completed (all of the zinc converted to $\ce{Zn^{2+}}$ and/or all of the $\ce{Cu^{2+}}$ deposited as copper onto the copper electrode. Charge separation does not occur. Then how do the charges remain balanced, you might ask. And that's where the salt bridge comes into the picture.

A Salt Bridge is a tube that connects the two half-cells to maintain electrical neutrality. This tube makes sure that the ionic solutions on both sides are electrically connected, but at the same time prevents diffusion of both solutions to become one (see the link to know exactly how this works).

Feel free to comment if you need any help or clarifications.