In the below diagram, a charge separation in the battery causes an electric field. When the wire is connected between the 2 terminals, what is causing the current to flow? Is it the electrons from the battery adding to the electrons in the wire, or the electric field within the battery (caused by a charge separation)?
A conductive wire is composed of atoms that have one or more outer electrons that are loosely bound to it. So it's "easy" for the outer electrons to walk from one atom to another: when an electron leaves an atom, a "hole" appears which needs to be filled by another electron (else the atom would become negatively charged).
When the wire is not connected to the battery, loosely bound electrons move in a "kind of random" way, from atom to atom, so the sum of all displacements is zero (hence there is no current in the wire).
When the wire is connected to the charged battery, on the right of the wire there is a deficit of electrons, meaning "a lot of holes to fill", and on the left of the wire, there is a surplus of electrons, and no "hole to fill".
So if we zoom-in on the left of the wire: if an atom A0 of the wire loses an electron, this electron will very likely go "to the right" because on the left there is no hole to fill. And the electron filling the hole of A0 will "very likely" come from the left (the battery), because on that side there are a lot more free electrons than on the other side. So this is how electrons from the battery "enter" the wire.
Now if we zoom-in on the right of the wire, we see the opposite effect, electrons leave the wire: if an atom loses an electron, this electron needs to fill a hole, and since the battery has a lot of holes, it's more likely that the electron will go to the side of the battery. And the electron replacing that electron will likely come from the wire (the left) because on the right there is a deficit of free electrons.
This is how electrons start to flow through the wire, because the sum of electron movements is now non-zero.
Do electrons exert a force on one another causing them to move like a chain of marbles?
I don't like this analogy because there is no "chain of electrons": the diameter of a wire is so huge compared to the size of an electron... It's more a "flow" of electrons.
Does the resistor slow down electrons by collision and therefore the distance separation between each electron now is greater leading to a decrease in repulsive force?
Not really, in a resistor, the outer electrons of the atoms can move from one atom to another, but they are more bound to the atom than in a conductor, so the probability that they will move is smaller. Hence, a resistor slows down the flow of electrons.