To continue to use your ball analogy think of the ball as analogous to the electron. Now what if the ball were attached to a point by a spring? Would it still fall? It can oscillate about that point but it would not be able to escape the restraining effect of the spring entirely. The same is the case with bound electrons. They are more or less bound to the atom. If the gravitational field is very strong it may be able t o break the spring and rip the ball out of the spring. This happens sometimes in electricity too. In a lightning discharge, the electric field is so high that even the bound electrons are ripped out of their atoms thus ionizing the gas and creating what is known as plasma. With a pool of free electrons and positive ions available electric current can now flow freely through the plasma - you wouldn't need wires. But unless you have a high enough electric field to produce ionization(for ionization of air the field required is close to $10^6V/m$ - such high fields cannot be produced by the 100 - 250 V household voltages available in most countries) you would have to use wires made of conducting material where free electrons are readily available if you want to have electric conduction at normal voltages.
It doesn't change. Whatever force acts on a positive test charge; the exact opposite acts on a negative charge (like an electron).
Any positive charge will attract the electrons, while any negative charge will repel them. Electrons therefore go from low potential states to higher potential states. This doesn't mean they're going from lower potential energy to higher potential energy! You may be knowing that electric potential energy is: $U = K\frac{qQ}{r}$. Since electrons have negative charges, putting in their value into the equation for potential energy, and changing the distance factor will show you that the closer the electrons are to the positive $Q$ charge, the lower their potential energy.
The electric field in a wire is from the positive terminal of the battery to its negative terminal. Electric field, if you remember, is the force experienced by a charge:
$$\vec{E}= \frac{\vec{F}}{q} \implies \vec{F} = q\vec{E}$$
When the charge is positive, the force will be in the direction of the electric field (note the vector sign). When it is negative, it will be in the opposite direction:
For an electron:
$$\vec{F} = (-e)\vec{E}$$
which, as is visible, will accelerate electrons in the opposite direction as that of the current.
$P.S:$ Current in a conductor is due to electrons only. Protons don't move very freely, since they're bound to nuclei.
Best Answer
The metals conduct electricity because they have a high concentration of "free" electrons in them. These free electrons exists even in absence of current, they don't need to be sent or injected into the metal. Water does not have free electrons. All electrons are bound to water molecules or to $OH^-$ ions. In electrolites there are other ions but still no free electrons.
The reason for some materials having or not having free electrons (condoctors versus insulators) resides in both chemistry and the structure of the material. It's not a simple function of just one parameter (see diamond versus graphite conductivity).
If you somehow "inject" electrons into an insulator (like water) they will not make it conductor. They may attach to the ions already present or otherwise interact with the molecules of the insulator, creating a static charge which eventually will discharge into the air or surrounding bodies.
And lastly, the "pushing" between electrons is not relevant to electrical conduction/ Actually it can be neglected. After all this is why we can get a basic model of conduction based on free electron gas assumption. The interactions that matter are the interaction of the electrons with the electric field (created due to the power source) and with the lattice of positive ions (one of the mechanisms responsible for the electrical resistance).
Edited to answer extra concerns:
[1] "Free" electrons can move through the lattice for both conductors and semiconductors. The mean free path is much larger than the distance between ions. [2]In free electron conduction the electrons don't jump from atom to atom and they don't need any vacancies. You have to understand that electric current is a collective phenomena. Electrons don't move one by one, pushing each other. [3] Superconductor may not require but conductors do. It's not a matter of interpretation. To start a current in a superconductor you still need an electric field.