No reputable body piercing place will pierce anyone under 16, period. There are regulations against it. Additionally, most will not touch a pregnant woman due to possible licensing implications. I get all my work done at Staircase, I've never been to the other places and while I'm certain they're clean and use an autoclave, etc. - I would be very concerned about them piercing a 5yo.
Personally, I am heavily pierced and inked and I wouldn't take my child to a piercing shop to have a needle put in and go through that experience. I would wait until my kiddo is 16 and let him/her decide on the jewelry when they're old enough to take care of it. They won't put studs in your daughter's ears, she'll end up wearing some CBR. I'd hold off, your mind is in absolutely the right place but she's too young.
I have a 10yo stepson and a 3yo son, my 3yo wants tattoos like mommy - I told him I'll take him when he's old enough, but for now we can do fake tattoos :) He wears my larger rings over his ears and we get creative with my plugs (9/16" lobes).