I think it's pretty harmless. My sister had it at that kind of age and mainly just spent her time decorating houses and dressing her Sims in different outfits. So basically like playing with animated dolls.
Yes, it will be fine - the game is pretty family friendly. You can make them have sex though (to get pregnant) but you can't see anything, they disappear under the covers!
Be careful though if she wants to download things for it - stick to major sites like modthesims.info, because there are some more adult downloads for it. (Mod the sims is moderated so all family-friendly). TBH most sites which have adult content will have big warnings stating you must be over 18.
I'd guess she wants the sims 2 or 3, whichever version her friends have, but unless your computer is very new it might struggle playing the sims 3. This is a helpful guide
Basically, you create a little person. You can choose what they look like and you can decorate their house as you wish, if you have the money. Then the little people have to find a job, feed themselves (learn to cook etc), take care of their house and make friends. If they make very good friends with someone it may turn into a romantic relationship in which case that person could eventually move in with your Sim and they could get married. They could also have a baby, pets, holidays etc depending on the choices you make and the extra packages you buy. It's all very very harmless. There's no nudity or swearing or anything else I can think of that would be unsuitable for a small child. I'd be perfectly happy for my four year old to play it, actually. Although at four she would probably neglect her Sims and I'd feel bad for them.
DS (11) has Sims2, which is rated 12+. It shows two people getting under the covers and lots of ... er, activity under the covers. DC all think it's hilarious, they know what that activity is about, they can't even be arsed to be embarrassed. Nevertheless, I'm not sure I would have let DS have it had I known. The people can walk around nearly nude (down to a bra and pants? I think, bikini at least), too.
It's quite annoying if they neglect the baby, too (lots of crying). Still better than shoot-em games, I admit.
6 months old should probably be ok, but I can check for you if you can post your system specs. I assume you're using Windows Vista? Click on the start "orb" and type dxdiag into the start search bar, press enter, copy and paste what it says here. (In Windows XP it's the same except "run" not "search")