Sorry to hear your dramas with the phone.
I have mine now over a month.
I use a flip leather case now, but had none for the first two weeks and I use my phone a lot but with no screen protector and I don't have any scratches on it.
You must have hit something with sharp edges hard.
I had a SG2, never had a screen protector and was exposed as I didn't use a flip case. It never got any major scrathes, but hundreds of tiny ones that could be scene in the right light only.
This is a much better glass in my opinion.
Hope you are able to fix yours without too much trouble.
Phone: HTC One - ROM: Android Revolution HD v31.5
Firmware: 3.57.401.500, Hboot: 1.55, S-OFF, Recovery: TWRP 2.6.3,
Radio 4A.20.3263.30_10.38l.1157.04L, Kernel: Stock 4.3
Tablet: Samsung Galaxy Note 10.1 - 16GB 3G N8000 - External SD Card 64GB
ROM: Android Revolution HD v10.0 - Android 4.1.2