Hi all, sorry if these have been answered before. Had a look on google and search bar and didn't really find what I'm looking for.

First off, I have a Les Paul copy. I'm having problems with tuning stability and want to install a graphite nut - I've also worn the original down a bit by switching from 11s to 9s.

I saw some precut ones in rockers in denmark street the other day so I'll pick up one of those.

How would one go about removing the old nut and installing a new one? (In detail if possible, I'm a bit f*cking of a spazz when it comes to DIY)

Secondly, I've been having problems with my Korg GA30 tuner. I've had it about 5 years, and it can't recognise the pitches correctly anymore, so having tuned I then have to fine tune by hand because E on the tuner isn't E at all.

2.) Can anyone recommend me a decent tuner that actually recognised pitches correctly and isn't too expensive? I've tried a QuickTune one, but thought that it sucked.

Thirdly, I've been having a problem intonating my guitar, in that the 12th fret of the low E string is sharp in comparison to its harmonic, but I can't move the saddle back any further.

Any suggestions?

Any insight would be appreciated!
here's for the nut:

for the tuner, try replacing the batteries. and there's no 100% accurate tuner. when you tune, select your neck pickup then put the tone knob to zero. pick on the 12th fret when you tune.
here's a good tuner:
Ibanez RG321MH (Air Classic/Tone Zone)
Fernandes Telecaster (Twang King/stock bridge pickup)
Blackstar HT-20 (Scumback 55 speaker/ Tung Sol tubes)
TC Electronic Nova Repeater
Lava Cables Clear Connect, Soar and Mini ELC
As a tuner I use my old, rather cheap multieffect, I don´t know if these utilities exist just as tuner, but I´d choose a digital one, it works really good for years now.
With tuners which are just tuners and are so small to put them in a pocket I´ve had problems since i play guitar.

EDIT: The multieffect is a Zoom GFX 1, though I doubt it exists anymore.