LOL! I'm the same .. bit of a clutz(I think) .. not sure.
With my D300 I got a Katzeye for it. Removed old screen, put it on cloth and noticed a large hair like mark on it .. thinking a hair. Nope a scratch.
There was no scratch on it when I took it out, When I remove mine, I use a pec pad on the mirror, and turn camera right side up so it falls gently on the pec pad. No damage there.
I know I didn't slip unclipping the retainer clip .. so how it got scratched is
They have a locating tab at one side of the rectangle. this is used so that it sits properly in the camera .. is a good place to use tweezers on too
Did you order a standard screen replacement or a 'better one'?
That's the one thing I keep forgetting to do, is to order a K-type screen for the D800 too .. maybe with some simple aids(microprism maybe .. not sure).
Last note on the hair in there annoyance. if you tried canned air, when you spray in the camera vf head area with screen out, are you holding camera upright(normally) or upside downly so that you can see it easily.
if upside downly(with top of camera pointing down) try it topside up .. as normal orientation.
Could be blowing hair around the vf chamber rather than out of it.
And of course the important reminder that the mirror surface is fragile and can be damaged easily so keep straw of canned air away as best you can.