I found that the strings are too distant from the fretboard. - > This adjustment is called "action".
Can it affect my playing, tone and whatnot? - > Playing yes, tone no. Shredding requires low action, and slide guitar requires high action.
I also found that the fretboard is too flat, almost like a acoustic guitar. Is it a problem or it's just normal? - > It's not a problem, it's just cheap. A radiused fretboard facilitates string bending and contributes to overall playability.
anyway i found that my guitar (Samick AV1) has an adjustable bridge. Does it mean that I can adjust the string distance to be closer to the fretboard? -> Yes.
How can I do it? Is there any specific method to do so? - > Yes. Take it to a pro. Let's just say that there's possibility your neck will have to be adjusted also. And let's just say cheap guitars generally don't respond well to adjustments.

But in your case, the RG is supposed to be flatter.
BTW, your guitar is still cheap.