Yes, piercings will close if you leave the earrings out. My second holes closed actually unintentionally. I ended up having to have them redone. I have two holes in each lobe and one on the top of my left ear. The double in the lobes in my opinion looks quite nice. I wear small diamonds in the upper hole in the lobe, and usually a small hoop or pearls in the lower. The one in the top of my ear I typically wear a small diamond. I will say, my mom didn't allow it, and it wasn't until I went to college and moved out that I had it done (no, I wasn't rebelling, it was just something I had wanted for a while). I am now 32 and still have it and still like it, so you may say that I'm a bit bias.
My honest opinion, it's your house, she's 13, and you make the rules. That's how I was raised. If you disapprove, then that's the bottom line. However, since (IMO) it's not a huge issue, and if you have any thought of possibly allowing it, then maybe it's something she has to earn with grades, etc, or maybe a sweet 16 gift. I agree that tons of piercings are gaudy, but I think a couple look nice... GL!