Do point charge really exist? Do point charge really exist experimentally? 
Am I right with definition of point charge? According to me point charge is a charge having 0 (zero) mass and have 0 (zero) volume.
 A: In your definition, you have put the mass $=0$ condition. You cannot have a massless charged particle. Permitting them to exist will predict a decay for electrons and we have no evidence that that happens. Of course, the assumption here is that QED is correct and we have no evidence that it is not.
However, if you drop the mass constraint then our humble electron should qualify for all practical purposes. As far as we know, the electron has no spatial extent. A spatial extent does not agree well with QED models.
P.S.:- I must clarify that throughout this post I mean electric charges only. There are other kinds of charges in nature as well (e.g., colour charge).
A: As of now, we've found no charged, mass-less particles, nor predicted any in the Standard Model. 
However, your definition of a point charge is flawed. Point particles have no volume: their spatial extent is similar to the mathematical idea of a point, regardless of their mass. Point charges have charge (and perhaps mass) with no physical extent.
Electrons appear to be point particles to the extent that we can measure them, both in roundness and size. Since they have charge, they (and their anti-matter partner the positron) satisfy most people's definition of point charges. 
The idea with point particles is that when the scale of an object is significantly smaller than either the separation distance or the apparatus, the point particle models the system well enough to make good predictions.
A: IMHO point particles do not exist, but maybe there's an alternative:
Let's start with one time and one space dimension. The one space dimension is, in fact, a two-dimensional structure, which we'll get from rolling up two-dimensional space very tight so that the radius of the cylinder will be in the order of the Planck length (which is pretty close to a one -dimensional space). We can place on this cylinder little circles, which represent the particles in this two-dimensional spacetime. Notice that cylinders can all be stacked on each other with zero separation difference, although the situation about the distance between two circles on top of each other is a bit more complicated, and involves the Planck distance.
This is all easy to imagine. But if we roll up a three-dimensional flat space in the fourth spatial dimension resulting in a two-dimensional (cylindrical) space which looks like a flat two-dimensional space, while it is actually a three-dimensional space, but because the Planck length is so little this won't be noticed, like rolling up a flat two-dimensional space in the third dimension resulting in an (apparent) one-dimensional space (which is, in fact, a two-dimensional cylinder which looks very much like a one-dimensional space because of its little radius of about the Planck length), we can't visualise anymore what happens due to the involvement of the fourth space dimension.
In this (apparent) two-dimensional space, the particles are two-dimensional spherical shells wrapped around the two-dimensional cylinder [like the one-dimensional spherical shells, AKA as circles, wrapped around the (almost) one-dimensional cylinder]. Also here the particles can touch each other "completely" like the circles on the very little cylinder.
Now we roll up flat five-dimensional space in the fourth space dimension. What results is an (apparent, but because the Planck length is...) three-dimensional space, but is, in fact, a four-dimensional cylinder on which three-dimensional spherical shells are placed to fulfill the part of particles. Again all these particles can completely touch each other [Picasso tried to catch the fourth dimension by painting his mostly female models, or imagination, by painting them from all sides (well, mostly only the side and the front view of the face together in one face)].
Notice that these structures have nothing to do with the strings and branes from which appear in string theory. These structures are rigid and can rotate without diminishing their length in the direction of motion. The angular velocity is perpendicular to the direction of motion, which is again easiest to visualize in the case of little circles moving on a very narrow cylinder. The objects can rotate faster than the speed of light without us noticing it. Maybe the thing we notice is that the particles have spin.
One more thing. In the rishon model created by Haim Harari, all quarks and leptons (and also the Z- and W-particles) consist of three more basic particles: the V- and the T-rishon. They are massless, but a combination of three of them, which have a colorless hypercharge and can have a colored charged like the quarks have, and a colorless combination like the leptons have (electrons, muons, tau particles), have mass due to the big strength of the hyper color force. So an electrically charged particle (or a color charged particle) without mass will not exist. Evey particle with mass has an electric charge or a color charge (but no hypercharge; they come in colorless combinations) and a mass without them doesn't exist (in the light of this model).
A: The question is ill-defined/slightly unscientific. 'Really exist' is somewhat metaphysical. Certainly, one may successfully describe experimental phenomena in electromagnetism and electroweak interactions with theories of charged, point particles. That's a theory invented by humans. Their actual existence, whatever that means, is an optional but unwarranted additional assumption, that is irrelevant to what can be described and what has been measured.
A: Definitely not. Point charges require stuffing a lot of charge into a very tiny volume, the electrostatic potential energy to do so would approach infinity. 
Electrons, though tiny, are not considered as point charges. Even they have volume, although quite small. See https://en.wikipedia.org/wiki/Classical_electron_radius for the value of radius of an electron.
