Let's start looking at your statements:
It was clear to me that all particles are merely excitation in fields.
This is not exactly true, even though often said. In the model of Quantum Field Theory, we describe an electron as an excitation of the electron field. It's important to stress, that this is just a description, i.e. it's hard to say whether it really "is" like that (whatever that may mean).
If electron is excitation in electron field, then how does it produce it's own field?
It doesn't. In QFT, the an electron field is postulated to exists everywhere (in our flat space-time). This is part of the theory base and as such, the theory can't explain it.
For whatever reason electron even after being an excitation produces an electric field, but why doesn't photon (or other such particles if there are) create any such field?
This statement is not completely true. In QFT, there is no electr(omagnet)ic field. In general physics, an electric field is the mediator of the electromagnetic interaction. In QFT that role is taken by the photon, i.e. an excitation of the photon field. These two fields (photon and electromagnetic) are descriptions in different theories but basically of the same effects. (Not precisely the same, but this is a bit harder to explain.)
This also answers your question as to why the photon (in qft) doesn't create such a field: because they live in different theories.
The fact that the photon doesn't couple to itself is basically an experimental fact. By now, we have a few nice descriptions and theoretical explanations of that, e.g. it's the gauge boson of an abelian gauge group or it simply doesn't have an interaction term in the quantum electrodynamics lagrangian. (The first implies the second reason, but they were found the other way round afaik.)
All these particles have mass and exert gravity how can they produce gravitational fields?
As far as I know, there is no consistent and tested theory that describes gravity and qft together. (more precise: ... results in them in a certain limit/under certain reasonable assumptions.) As Neuneck mentions in his comments, there are reasonable theories about qft in curved background.
In qft, mass is just a parameter of the theory nothing more. It's just our interpretation from other parts of physics, that gives it the meaning the mass we know.
Gravity is in this sense closer to normal electrodynamics. The gravitational field is our space-time and is bend through gravity-charges=masses. It's like putting an electron in empty space: First the electrmagnetic field is flat and 0 everywhere (casually said it's not there, but this is imprecise and can create confusions) and afterwards, yxou have a sink in the field and it's bent (casually said, there is a field).
when particles are excitation in some field how do fields arise from them?
With the answer above as background: They don't (in qft). It depends on the theory, what you mean with the word field. It's mostly a confusion because one doesn't always precisely define what one means with a certain word (like electron or field) in physics.