I was going to vote "No" until I read that it really bothered her. While 10 seems young, things are so different for our kids. Heck, I'm old enough that we still played with Barbie's in 6th grade w/o embarassment!
My mother wouldn't let me shave until I was 13 - and yes, I had dark, hairy legs. It bothered me, but back then everyone else was having the same battle with their mother - so no teasing went on.
Mom's reason was that we'd have to "shave the rest of your life - no point in starting already." (BTW - She was wrong, my hair now only grows in patches at the knee & ankle. Hooray!)
Unfortunately, kids always find something to tease about. I'm really big on removing the easy targets for teasing.
I never had much luck with electric razors. I use cheap disposables - but I
still cut myself when I open a new one so I tend to use them for months.
Re: the mustache - My good friend, who is a hairdresser, was actually shaving because she not only had the mustache, but also "beard" hairs. She'd gone all other routes first. Last year she underwent a laser treatments that killed the hairs. (Something about the light following the color of the hair to the route.) It was expensive, but it worked really well. I'd definitely consider for my daughter if she were having trouble so young.