Is it possible to create a completely new genre of music
This question might appear really weird, but still I ended up asking.
Its not weird at all. Many composers throughout history have asked this very question.
I will give a tentative "yes" to this portion of the question, with the clarification that by genre we are referring to the stylistic conventions of music, and not just a commercial marketing label as mentioned in the other answers. For more on this debate, see: What's the difference between genre and style?.
Granted, I could be wrong about this. Perhaps we have reached some kind of stylistic "saturation" where every possible stylistic choice possible has been made somewhere, and all music from now on will be doomed to repeat existing styles, but somehow I doubt it. All you have to do is look at music history to find composers contantly pushing the bounds of previous music and creating new styles of music; however, as you mentioned, this is done by modifying or mixing previous styles.
First of all, I would like to know some genres that appeared and took their inspiration from nothing already known at this moment.
I do not believe that is possible. Every new style is informed to some extent by some previous style. For this to not be the case, you'd have to strand a bunch of infants (whose pregnant mothers had been carefully isolated from all music) on a deserted island, and watch them develop their own style of music from scratch. It's an interesting thought experiment, and possibly relevant to a sci-fi author who has to construct a completely alien style music, but its not likely to ever occur in reality.
Just to illustrate the point in classical music: Atonalism grew out of the bold and unconventional harmonies that were introduced in the Romantic Era. The later were a reaction to the straightforward and formalized "Classical" harmonies, which themselves grew out from the rules governing Baroque counterpoint. This had been an extension of Renaissance polyphony, which was an elaboration of Medieval chanting, which was based (at least nominally) on Ancient Greek modes, whose origins have been lost to time. So while each of these was a huge step in music style and theory, it cannot be said to be completely unrelated to anything previously in existence. This is the nature of culture, as a sort of shared memory.
However, many new styles, when first created, are novel enough to be considered experimental, innovative, esoteric, or avant-garde. For example, Rite of Spring, though not technically atonal, was so foreign and bizarre that it apparently caused a riot at its premiere. Today, parts of it would not be terribly out of place in a film score. But it cannot be said to take inspiration from nothing preceding it.
And secondly, will it still be music as we know it today if we don't find some basic things related to one genre or another?
To some extent, this is going to depend on what you mean by "music". Is John Cage's 4'33 truly music? What about music improvised by elephants? For more on this type of debate, see:
What is the difference between a collection of sounds and music?
However, I feel that this question, as worded, contains an inherent logical self-paradox that makes any affirmative answer impossible. If we don't find anything at all that is "related" in some way to some type of existing music, then by definition it cannot be music "as we know it". However, it may well still be music as we don't know it.