How do electrons actually move in a circuit? Last year, we were taught about electricity, about how electrons move in a closed circuit. But as our teacher had not taught us about electric fields yet, she gave us a simplified model of motion of electrons in a circuit. (given below)

She told us that, due to chemical reactions taking place inside the dry cell, accumulation of electrons occurs at the -ve terminal and cations at the +ve terminal. When a conductor is connected with the battery, due to accumulation of e- at the -ve terminal, there is mutual pushing between electrons, and this causes them to move away from the -ve terminal and towards +ve terminal. Also, cations attract electrons towards the +ve terminal.
Q 1. Is this true that electrons are both attracted and repelled (by cations and other electrons respectively) in an electric circuit?Q 2. Is this analogy correct? What exactly is happening?
Q 3. Is this (below) a better analogy of electron motion (field lines)? Can electric fields be altered by conductors such as a metal wire? Will most of the field lines be inside of the conductor, Therefore providing a better path for electron flux/flow? 

This is not a Duplicate so please don't mark it as one. My real question is, are both analogies correct?
Similar: How does electricity flow in conductor when potential difference is applied?
Very similar : Why does electrical current start to flow?
 A: Your teacher's description is not bad.  The phrase about mutual pushing is vague.  I'm not sure if he or she means there is pushing to get things started, or pushing to maintain current, or something else.  I think it might be fair to say that mutual pushing establishes the charge distribution needed to maintain the current, which I'm about to describe.
Your picture is pretty good, too.  Once the current is established, charges accumulate on the surface of the wire in such a way that the surface charge density is positive near the positive battery terminal, negative near the negative battery terminal, and passes through zero somewhere in the middle.  The result of this gradient of surface charge is to induce a uniform electric field inside the wire, much as you have drawn.  It's this field that applies force to the charge carriers in the wire.  You might argue that the charge carriers will accelerate without bound (Newton's second law), but no, each carrier will eventually collide with an impurity or defect  and stop (or deflect, or turn back) the carrier, thus limiting the speed.  A thermal vibration can do the same.  Higher resistance materials have more impurities and defects, and thus lower average carrier speed.   Raising the temperature of the material increases the number of thermal vibrations and also raises the resistance.  This effect is prominent in a light bulb.
A: A1: Yes, that is true. However, and this gets hard to explain without quantum mechanics and other more complicated tools, a lot of things "average out" and make it so you can almost ignore them, or are kind of secondary effects. For example, in simple (but actually pretty effective!) models, you can assume that the electrons don't interact with each other besides colliding sometimes. In reality however, they are constantly interacting with everything in the material, which is why you make these simplifying models.
A2: Sorry, which analogy?
A3: It's somewhat like you drew, but actually the field lines mostly end up inside the conductor. It's not like you just draw the normal electric field lines between two points, and then the ones that happen to lie in in the wire in that drawing are the ones there -- they end up "taking" most of the electric field inside of it.
A: The battery does create an electric field near itself that forces electrons away from it. As they travel they begin to build up on the bends of the wire. This is known as feedback. The next electrons coming by are repelled away from this build up allowing them to travel around instead of just directly away from the battery. This is how electrons can travel through a wire even if you bend it 90 degrees like a square. In fact, electrons on the surface of the wire are what force other electrons inside the metal to flow. They begin with an excess of electrons near the negative end of the battery and slowly become a deficiency of electrons on the surface of the wire near the positive end of the battery.
Imagine the surface of the metal wire as having traffic control people pointing their hands where the electrons inside the metal should flow. Near the negative end they say keep it moving and point away from the battery. At a bend in the wire they stand at the bend and point away telling incoming electrons to make the turn. Then finally near the positive end they say come on down right this way, come toward me. Those traffic control people are built up charge on the surface of the wire. 
A: I would like to answer the question in few points that would clear few doubts.
1) The atoms of the conductors have valency electrons that are loosely bound to the nucleus. Also the crystal structure of the metal conductors have the sea of valency electrons which are free from nucleus bonds and are free to move upon external potential. So the electrons move all along the conductor when external field is applied.
2) The magnetic field lines are the hypothetical representation of the direction of force on an +ve charge at any point in the space, the tangential direction. It is not the path of the motion of the charges. Its not this field that causes most of the movement of the electrons but the electric field produced all along the conductor.
So your first representation is to some correct.
