I bought this cheap steel-string child's guitar because I moved and left my real guitars in my old city. I'm actually a petite girl so I love that it's such a small guitar, but the G, B and e strings don't play right. It generates the same note on frets 1 and 2. It's fine once I get up the neck. The high strings also rattles unless I am fretting and it sounds best higher up the neck.

Is there anyway I can fix this cheaply? I've read reviews where people said they filed down the bridge to bring the strings closer to the guitar and lower the action. I dunno. It was $40 and doesn't have a truss rod, so I'm thinking maybe I bought some cute wall art?
A conservative is a man who believes that nothing should be done for the first time. --Alfred E. Wiggam
I do believe you bought "cute wall art." My friend has a guitar similar to what you're saying, and we tried to make it playable, couldn't really find away... I recommend going to a pawn shop and looking at their guitars, sometimes you find a cheap one that's not horrible to play.
Epiphone Zakk Wylde Les Paul =D
Here's the problem.... Your kid's instrument might be able to be made playable, but unless you can do the work yourself, it will cost you more than the thing is worth.
If you're totally unfamiliar with making the basic adjustments to the action and such....

You could go to the Frets.com site and read the "basic instrument setup" section to see how far off things are. It's not just a matter of "filing down the bridge".
Hmmm.... yeah, you will end up paying more to fix it than you probably paid for it. Chances are, the neck is warped,, or needs some adjusting. But, from the sounds of it, it seems to be warped. I had the same problem with a Martin guitar which was a gift to me. Now, Martin is a great company. I don'y know what happened with this one, but I received it from my uncle, who got it from a neighbour who moved.The neighbor obviously didn't take care of it . The term used to describe what's happening to your guitar is fret buzz, and it sounds like its because of the neck being warped. But, you can get a guitar setup for about 50 bucks including the price of strings. If the neck is not warped, they should be able to fix it. If it is, just forget it. If it's a toy guitar, it's not worth the money required to fix a warped neck, which is in the hundreds. Good luck, hope you can get it fixed.
Happy Playing!