It's been a while since I studied that passage, but as best I can remember there's two different, and possibly 3 different words there being translated as "covering" (as well as "power" in v10). Some are a reference to hair, some to head covering.
Verse 10 is a reference to a Midrash based on Genesis chapter (5 or 6?, I'd have to look) where it says "sons of G-d saw the daughters of men...". According to the Midrash it was the women's hair that caused the "fallen angels" to go astray, so Paul is arguing that women should wear a head covering "because of the angels".
There's nothing in scripture that specifically forbids women cutting their hair. I am aware of denominations that interpret it that way, but the point Paul was making was more about women covering their hair rather than having long hair. Verse 8, as well as the last part of verse 7 is consistent with this, her hair is for her husband, not for the general public.
In Judaism it was and still is the custom (in Orthodox Judaism, less so in more liberal movements) for married women to cover their hair in public. This is one of those customs that was apparently present in early Christianity but was dropped a long time ago by most of Christianity.