Half a semitone is a lot to be out. First, are you sure you intonated at the 12th fret (using the 11th or 13th has been done before). The nut being high can mess things up a bit but not normally by that amount. And the frets would have to be a long way out of position to be that far out, and they don't look it.

But the kit strings are utter rubbish and of variable thickness and output and are only good for basic set-up like action and nut slot depth when you are taking them on and off a lot and it saves spoiling a good set of strings. Always do the intonation with a good set of strings fitted.

To adjust the nut, you either need a set of nut files, or you need to remove the nut and sand the bottom down and put it back again. (and repeat until the nut is at the right height). Good nut files are expensive (a worthwhile investment if you work on several guitars though), so filing down the bottom of the nut is the normal route without them.

Tapping lightly from either end using a hammer and a largish flat blade screwdriver is my normal method of nut removal. they should be help in by a smear of PVA glue which stick well to the wood but not the plastic. You can also tap lightly from the headstock side of the nut, but don't tap towards the headstock as it's very easy to break off the end of the fretboard holding the nut in place.

You'll need to clean out the old glue from the slot. I normally use a mixture of a sharp blade and a needle file.

I wouldn't glue the nut back in place until you've done all your set-up adjustments and are 100% sure the nut is the right height. String tension will keep it in place and I normally loosen the strings off, smear the nut bottom and the slot sides with a little PVA or Titebond, slide the nut back in press it down, remove excess glue with a wet cloth or paper towel, then replace and tighten the strings to clamp the nut in place whilst it dries. Just make sure the nut is centred properly so it's not sticking out of the slot on one side of the other - but you should find that the strings centre the nut on their own.