I was tuning down, and my high e has just snapped. Also, just now, I have fretbuzz at the 10th fret on my D string and 8th fret on A string. Why has this happened and how can I stop it?
"Tuning down?" To what? If your strings are too slack, then you are quite likely to get buzz.

If you have an electric with an adjustable bridge, then tweaking the action at the bridge up a bit might take care of the problem.
Buzz is normally the result of the vibrating string hitting a fret.
If you have a floating bridge of course all the strings are going to buzz if a string is broken. Your bridge is not in it's zero position because of the missing string, which in turn is causing the action to be too low, and all the strings will buzz because of this. Replace the broken string.