Wow. Pretty divided on this.
My mom made me wait until I was 12 to get my ears pierced. When my daughter tells me she wants her ears done, that means she's ready. I'll make sure that's she's mature enough to keep them clean and know the consequences of not taking care of them.
I actually worked at Claire's in my early 20's
I wouldn't hesitate to bring her there, unless it's some 17 year old, I might ask when someone older is working. It's not unsanitary because the only thing that goes through the ear is the sterile earring. Merle Norman is an even better idea. Great ladies there.
I would say 10 years and under, get them done at the same time. Less chance of walking around with a single.
I also would have no issues with taking her to a tattoo place, but only if she's older, because I don't think they could do both at once. And a needle isn't as quick as a gun, so I wouldn't want to see her in too much pain at a younger age...
Go with the regular sized studs vs the mini's because of possible (slight) chance of complications (the ear grows over the earring and the earring gets embedded in the ear) If someone tells you to clean it with alcohol, turn around and walk the other way. If your daughter has long hair, keep an eye on the back.. hair gets wrapped around the shaft behind the backing and "ear gunkies" gets stuck to it and it gets stinky!
To the person with the daughter with grown over holes; after your daughter having a long hot shower, try putting the earring in through the back of the ear... sometimes I've had better luck going that way.
Good luck!