I'll try to Adress all your individual question. There are many, and it could be fruitful to split the question into two. But here we go.
Now how this potential difference is created
The battery potential difference is created by the internal electrochemistry which forces electrons to one pole. With many electrons accumulated there, this pole will have a higher electric repulsion than the other pole against any additional electrons arriving.
I mean there must be some constant (or not constant?) electric field, along/against which a charge had been moved so as to increase potential energy. So how this occurs? potential difference without field how ??
Indeed, and that is in fact an internal electric field caused by the chemical potential within. This field causes an electric force that forces electrons to accumulate at one pole. At some point the accumulated repulsion balances out this internal force - and then the battery is fully charged. The internal chemistry is in your example designed to exactly provide up to a 2V potential difference.
so why not the mobile charge due to which potential energy is there in the battery, move and thus causing self depletion
Because the internal electric forces caused by the chemical potentials are stronger than (or exactly equal to) the accumulated electric force that tries to make the electrons move away from the pole. Also, they can't jump out into the air since the air is not conductive and thus resists them even more strongly. They are thus stuck on that pole until a conductive path is provided.
Taking one terminal as high potential and other as low (which one is the one ?)
By default we always refer directions and highs vs lows to how a positive charge would behave. With many electrons accumulated on one pole, making it negatively charged, then if a positive charge was placed nearby, it would want to move towards this negative pole. Since objects always want to move towards states of less stored energy, then the negative pole is thus the one with low potential and the other and comparatively positive pole is associated with high potential.
Note, though, this is only a default terminology. A book that specifies that it is concerned with the energies of electrons as charge-carriers might use the terms oppositely, since an electron will want to move away from the electron-full pole, thus associating that pole with a high potential from the electron perspective. So be clear on the context in various textbooks.
electric field should go from one terminal to other terminal. So how electrons in the wire start moving due to field ?? How this field bends along with wire?
They move because they "queue up". An electron at the negative pole wants to move as far away from the pole as possible due to the enormous repulsion. When the only conductive path is away from the pole, then that is the way it will move.
This electron would then theoretically be satisfied with its new location somewhere within the wire and would not want to move further. But then another electron arrives. And yet another. Etc. Each electron carries its own electric field with it, and when queuing up like this, their electric fields will again accumulate like on the pole. But the accumulate will be momentary since the first electron of course simply will move further in on the wire.
Eventually it will be pushed all the way through. Simply because it isn't alone in the circuit. And at any bends or corners in the wire, there will in the same way we a slight tendency for charge accumulation until a charge inevitably will find the new path to move along.
Is this field constant (why)
It might be, for typical batteries that aren't depleted, yes. The queuing up of electrons effective means that any push on one is passed on to its neighbour. In this way the push is eventually the same throughout all parts of a (series connected) wire.
As long as the battery can sustain the constant 2V at the pole, that is - the current flow away from that pole thus mustn't be quicker than the chemical activity that moves electrons to this pole. This is typically labeled on a battery as the possible maximum current that can be drawn from the battery.
what happens to the field inside of the battery
It is upheld for as long as the chemical potential exists. The mechanisms here are very technology dependent and could for instant depend on the ratio of acid molecules to the electrode surface atoms.
How energy is supplied to electrical appliances in the circuit? In what from electrons carry energy?
What we call electric energy is essentially kinetic energy of the charges. That is, kinetic energy at the sub-atomic scale from the violent fluctuations and vibrations which particles like electrons constsly undergo. When electrons are slowly pushed through the circuit, then suddenly they are pushed into, say, more dense materials like the filament of a light bulb. This is also a conductor but one with higher resistance, meaning the electrons will jump into the material atoms.
With a collision happens energy transfer. Kinetic energy of the electron is passed on to a material atom within the material lattice. Eventually many atoms have now gained kinetic energy and are thus moving in rapid vibration. This is what we defined as thermal energy and temperature, and this eventually might glow and thus send off photons - then electric energy is converted into light.
why this energy is not dissipated in the wire (since they are travelling against electric field)
The energy that electrons carry is to some extend dissipated in the wires via collisions with the wire atoms. Wires will indeed get warm as well. But they are typically optimised to be as little resistive as possible to avoid this energy loss.
I hope this helps. Please let me know in comments if something is unclear.
