You've asked some really good questions here. Before starting, I want to first mention that the traditional picture of particles moving through a wire in electostatics is missing some physics; for instance, it ignores the quantum mechanical nature of electrons. The reason we still teach this model is because it captures the main effects (the phenomenon of current) without dealing with microscopic details, but I wanted to warn you that some of the answers will involve physics that is probably not contained in your readings in electrostatics.
To put things in perspective, we now know Newtonian physics is "wrong" (or perhaps more accurately, incomplete), and doesn't give the right answers if, for instance, an object is very small or moving very fast. But we still teach Newtonian physics because it's "good enough" for describing macroscopic objects like cars and baseballs.
Now, to answer your questions,
When electrons start moving through the wire to the positive terminal, do they
all move at once? Because otherwise, while they are moving, they will still
exert repulsive forces on each other? Does this repulsive force affect their
movement?
The microscopic picture of a metal is (crudely) a collection of negative charges, aka electrons, moving through a lattice of positive ions. Indeed, there will be an attraction between these ions and the electrons, and repulsion between any two electrons. Surprisingly, there is also an attractive force between the electrons. The origin of this attractive force is that the electrons attract positive charges around them, and can in some cases lead to the formation of a bound state called a Cooper pair, which are relevant for explaining the phenomenon of super-conductivity, a phase of metals where the resistance is exactly zero. Note, this requires quantum mechanics to do properly, and is extremely subtle.
Shouldn't some of the electrons stay in the wire itself? If, at some point of the
wire, there is not enough repulsive force present, will they stop at all, or
will they reach the positive terminal?
Again, we need a more refined model, in this case statistical mechanics. Before connecting the terminals, the electrons all have a random distribution of energy which manifests itself as temperature. The presence of an electrostatic field causes a net flow of charge, but at the micro level, electrons are colliding and moving in a variety of directions. Often times you will see electrostatics books speak of drift velocity of the electrons, which is a statistical representation of the net flow. A single electron is probably moving much faster than the drift velocity, even perhaps in the opposite direction of the current flow, due to the random thermal energy and the collisions between particles.
Will the shape effect the movement of current? Does it have any effect on the electric field?
In electrostatics, no, but in reality, yes. In mechanics, one has statics and dynamics. In electromagnetism, one has electrostatics and electrodynamics. If you keep learning about electromagnetism, you will soon encounter another field, the magnetic field, and you will learn that the electric fields and magnetic fields are intertwined in such a way that lead you to reconsider the two fields as components of a single entity (hence, "electromagnetism"). In particular, you will learn that current carrying wires produce magnetic fields (Ampère's Law) and that changing magnetic fields can produce EMFs (Faraday's Law). This is a legitimate concern for building real world circuits, and the quantity associated with this effect is called impedance. Impedance is measured in Ohms, like resistance, and depends on the geometry of the circuit.
Will the length of the wire effect the speed of the flow of charges? If we have an infinite length of wire, will charges flow at all?
You're definitely on to something here. The resistance of the wire is proportional to the length of the wire. By Ohm's Law, the current is inversely proportional. The current is proportional to the drift velocity, so the current is inversely proportional to the length of the wire. See http://hyperphysics.phy-astr.gsu.edu/hbase/electric/ohmmic.html#c1 for a derivation.
Best Answer
We absolutely observe this in daily life. Every time you see a motor running, you see the effect of the force of currents on each other. I recommend the following experiment:
Create an apparatus with two parallel conductors (fairly close together) where you can adjust the tension in the wires (like a guitar - but make sure you take care of insulation). "Tune" the tension in the wires to twice the frequency of the mains power in your country (50 Hz or 60 Hz). Then make the wires part of an electrical circuit. You will see them starting to vibrate - confirming that there is a force (with a frequency equal to the frequency of the mains power) between the wires. Why twice? Because if the currents in the two wires are in phase, the direction of force will be the same twice per cycle - so a 50 Hz mains frequency will excite a string tuned to 100 Hz.
Usually, conductors run in pairs, with a fixed distance (insulator) between them. Such a configuration prevents you from noticing the force between them. Note also that the force is not large - $2\times 10^{-7}~\rm{N}$ for two wires that are 1 m apart and carrying 1 A of current. Which makes it hard to notice unless the currents are very large, or you set up a sensitive experiment.