Re: How do you stretch strings?
+1 to all of what Blueman said.
Wipe the strings down with naptha to remove any machine/finish oil from them.
Don't over-stretch your strings, you can "deaden" them by doing so.
I like to use a pick flat under each string and once I get it up to tune, slide the pick from the bridge to the nut, once, as far as I can at the the height of my finger. IOW, the pick is between my finger and the string.
For lube, I've used graphite, vasoline, chapstick and currently powdered graphite mixed into vasoline.
If after all this, you hear any pinging as bring your strings up to tune, your nut needs a little work.
If you feel comfortable doing this, get some 500/600 grit wet/dry sandpaper, fold it over (once for the unwound strings, twice for the wound strings) and
VERY GENTLY rub in the nut slots, arcing the paper from side to side(like a sail boat bobbing on the waves)... no more than 10-15 swipes.
Keep the sandpaper angled down toward the headstock at a slightly steeper angle than the tilt of the headstock or the string angle if you're on a Fender.
You can "flare" the headstock end of the nut slot making it slightly wider than the slot.
Then do this again with 1500/1600 grit wet/dry sandpaper. This will polish the slot, again gently, but you can use a little more pressure... again about 10-15 strokes.
Be very careful that you don't increase the depth of the slot.
I find that adding the graphite/vasoline mixture into the slot and then wiping most of it off gives me enough of a visual indicator of how deep I'm going.