So I'm kind of having a hard time playing on the higher frets with my guitar as the string height is a bit too high for me, so I guess that my question is, how can I solve this (if possible at all)?

My guitar has a neck-thru-body 25.5'' neck with Extra Jumbo frets. It's also equipped with a floyd rose special bridge.

More info about the guitar:

I'm currently using (11 - 15 - 22p - 30 - 42 - 54) strings (Ernie Ball: Beefy Slinky) with drop C tuning.

So I want a lower string height, however, if I lower the bridge I get a buzzing sound from the strings, so that doesn't seem to help that much.

Help would be greatly appreciated, as I'm kind of having a hard time playing guitar solos and stuff like that.
Truss rod may need tightening. If you don't feel comfortable maybe you need to have it looked at by someone. Anyone can tighten a truss rod, the key is not to over-tighten it, and keep track of how far you turn it so you can go back if necessary. For the record, you tighten it by turning it counter clockwise. If the neck is bowed, tightening it will also lower the action which is what you seek. Good luck.
Press a string down at the 1st fret and the 24th fret, then look at the 7th fret. If there's a massive gap between the top of the fret wire and the string, you have too much neck relief and need to tighten your truss rod.

I'm going to guess that your guitar didn't come from the factory/store with those strings, so you've put them on without adjusting the truss rod to compensate for the extra tension.