Why the EMF of a battery doesn't depend on distance between the two electrodes? I have read that resistance of a conductor is directly proportional to its length. So, the EMF of a conductor (electrolyte of battery) should increase with increase in length/distance between the electrodes..is it not so?
 A: I'm not going to use too much "Professional Terms" here. I'll just give you a rough idea on what's going on.
EMF is measured in volts.
Volts stands for work done to bring a unit charge from infinity to a point.  
You can think of Volt as the "Apparent Charge Density". Similar charges repel each other, the "apparent charge density" is the force of repel ( or attract ) a charge feels as it gets nearer to another charge.
The higher the volt of an electrode, the higher the "apparent charge density" and you need more energy to bring a charge near that electrode ( for repelling charge )  
With that said, EMF is the ability for the Cell to push charges and charges together even though the charges are repelling each other.  
You need to understand:
 1. Electrodes ( any simple piece of conductor ) can store charges if there is voltage applied to it. So the Electrode acts like a Capacitor.
 2. The higher the voltage applied, more charge can be stored on the Electrode.
 3. When charges are stored on the Electrode, the Electrode will produce an opposing Voltage to prevent further charges from flowing into that Electrode. ( Similar charges repel )
 4. If a Cell's EMF is higher, it will be able to push more charges into the Electrode.
 5. If an Electrode is bigger in volume, it will be able to store more charges before the opposing Voltage rises high enough to completely prevent charges from the Cell to flow in. ( More Charges per Volt )
With That said, having a larger electrode will not decrease the EMF as long as the stored charges are not removed ( no charges flows from one electrode to the other ) because the Cell will still pump charges into the electrode until the electrode produces a strong enough opposing voltage to stop the cell.
So measuring the ends of the Electrodes ( Cathode and Anode ) will always give the same amount of EMF ( and it is the value of the EMF of the Cell ) regardless of how many resistors you connect between your measuring device and the Electrodes as long as no charges flow between the Electrodes while you are measuring.
Please be aware that commercial Voltmeters actually allow small amount of Current to flow in them, so they cannot measure the exact EMF of the cell. And if your Electrode's Resistance is higher than the Resistance of the Voltmeter, you will get a different reading of EMF since charges are flowing away faster than the Cell can pump in.
A: If you are talking about the open circuit voltage of the battery, then internal resistance and resistance of any conductors connected to the battery are irrelevant.  The voltage across a resistor is proportional to the current thru it, see Ohm's law.  When there is no current, there is no voltage drop across a resistor, so both ends are at the same voltage.
Battery voltage does go down with current because batteries have internal resistance.  The resistance of the wire to the battery is usually very small compared to the internal resistance of the battery, so except for very high current applications (like starting a car), the resistance of the wires can be ignored.
Bigger batteries usually have less internal resistance (more stuff in parallel), and are therefore capable of higher current.  Internal resistance does go up with distance between electrodes, you you can't confuse that with distance between the external contacts of the battery.  Round single cells, like the common AA for example, a electrodes and electrolyte rolled up.  A D cell has less internal resistance that a AA cell of the same type because the overall flattened cell area is larger.  The distance between electrodes is still the same, there is just more area all rolled up.
