Does the Hawking Radiation give off X-rays?
We have no evidence of that. Nor do we have any evidence that Hawking radiation takes the form of gamma waves, or light waves, or radio waves, or any electromagnetic waves of any frequency, or anything else. We have no actual evidence of Hawking radiation at all. You may have heard of analogues being offered as evidence as per this physics world article. But I'm afraid they aren't evidence, because they do not correctly model a black hole.
Mostly I read that the x-rays are produced from the matter surrounding the black hole which spirals into the black hole due to its gravity and is consequently heated up. If the black hole has a main sequence star or a giant star as a binary partner, the BH pulls matter from the partner, which then forms an accretion disk around the black hole.
I think most people are fairly happy with radiating black hole accretion disks. I am. It makes logical sense, and we have what looks like good supporting evidence of that:
Image credit: Gabriel Perez Diaz, Instituto de Astrofisica de Canarias (Servicio MultiMedia)
But sometimes I read that Hawking radiation is also producing X-rays or even gamma rays.
Yes, but I'm afraid such articles are speculative. As I said above, we have no evidence of any Hawking radiation of any frequency. You can read a lot of other speculative things about Hawking radiation. For example see Wikipedia:
"Physical insight into the process may be gained by imagining that particle–antiparticle radiation is emitted from just beyond the event horizon. This radiation does not come directly from the black hole itself, but rather is a result of virtual particles being "boosted" by the black hole's gravitation into becoming real particles. As the particle–antiparticle pair was produced by the black hole's gravitational energy, the escape of one of the particles lowers the mass of the black hole".
This is lies-to-children I'm afraid. Virtual particles only exist in the mathematics of the model. They aren't real particles, they're field quanta. It's like you take a field and divide it up into abstract portions, and say each is a virtual particle. There is no magical mysterious way to "boost" some abstract portion of a field into a particle-antiparticle pair. Wikipedia also offers this:
"An alternative view of the process is that vacuum fluctuations cause a particle–antiparticle pair to appear close to the event horizon of a black hole. One of the pair falls into the black hole while the other escapes. In order to preserve total energy, the particle that fell into the black hole must have had a negative energy (with respect to an observer far away from the black hole)".
This is yet more lies-to-children I'm afraid. Vacuum fluctuations are real, being the electromagnetic equivalent of the little ripples you can see on the surface of the sea. But they are not the same thing as virtual particles. And just as there are no pencils less than 0cm long, there are no negative-energy particles.
So is this possible or very rare and only occurring within very old BH?
I certainly think it's possible for a black hole to produce x rays and gamma rays and light waves and radio waves. But as to whether these are produced by Hawking radiation, I have to say that I don't know of any mechanism by which this could happen. John Rennie referred to another question here, which he attempted to answer. Arnold Neumaier attempted to answer too. Note this: "As everyone using such fairy tales, he says nothing at all about how the fairy tale could be grounded in real physics, and hence why it should contribute to understanding".
Or is Hawking radiation only about virtual particles which often annihilates each other or one going into the BH and the other becoming real?
It definitely isn't about that. If particle pairs were being created above the event horizon, the chances are that they'd both fall into the black hole. The black hole would thereby be consuming the surrounding vacuum energy, and growing rather than shrinking. I'm afraid what's it's really about, is that Hawking radiation is a speculation that's been around for so long that people rather take it for granted.