Hio

I always have trouble with my G-String not being able to be tuned. Every time the string is being tuned, it must be getting stuck in the nut because the pitch doesn't change then suddenly there is a little twang noise and the string is higher in pitch than I need it to be. Then I tune down towards the open G note again and changes pitch till I get to the right tuning. I start to play and I find that it has already gone flat and the cycle starts again.

The other problem I am having with the string is that when I play an open note, the string is "roughly" in tune but when I play the note on the 2nd fret, the A note being played is sharp by about half a step. I have tried setting the intonation but the saddles on my SG will not go back far enough to set the intonation correctly. Is it possible that the strings I have chosen are too light for this model of guitar? and consequently the intonation can't be set for strings so light?

I have been forced to tune the string to the open G note, then tune it to the 2nd fret A note. When I play the open G note, it's flat but it doesn't sound that bad and when I play the 2nd fret A note, it is still a little sharp (because I didn't tune it exactly) but doesn't sound that bad either. Still.. the guitar is technically out of tune.

What's going on?