SKIN PROBLEM!

Very col subject isn'it?So after a period of time at the edge of my fingers growing up a fat tough skin..And when i play something my fingers stuck at the frets..Should i remove it somehow?Or not?

what you have their are callouses. That's what allows you to play without going in some amount of pain. I don't know how your fingers get stuck though...
I personally had a problem with those callouses, since I play the guitar for 6 hours a day, 5 days a week. They grew, and after a while became so thick that they kind of peeled off, so for a couple of days, I had to play with my bare fingers, and it hurt like hell. So a doctor gave me a medication that's used for hunting dogs' paws, that toughens your skin almost immediately (in a couple of hours actually).
Doing the dishes, wearing skiing gloves, taking a long bath can hurt your callouses. If yours are weird, I guess you could use some very thin sand paper to smooth them out.

Whenever I cut my fingernails (which I do pretty frequently on my left hand) I take a metal fingernail file (I hate ebony boards) and smooth the edges that the clippers make. After I do this, I file my callouses, so that they smooth out with the rest of my fingertip. This prevents the place where your callous meets with the uncalloused part of your fingertip, where the callous may poke up a bit, from catching on frets when sliding or just moving your fingers near the fretboard. Filing and then applying a little bit of hand lotion is a great way to have soft, smooth skin, but you still won't hurt when you play. I also personally file my knuckles. It's just like the kind of training where people hit metal or wood over and over again to strengthen their hands, but without much work. Once again, though, APPLY LOTION after filing. If you don't, your skin will become dry and hurt more.