There are two different types of DS. The original one is going for about $130 I believe, and comes in silver, blue or red (red if you buy it in the special Mario Kart package). The other one, the DS Lite, is smaller and has a brighter screen, among other changes. That sells for $130 also, and comes in pink, black or white. As for your questions:
1. Like I said, probably around $130 for the system itself. Check your local stores to see if they are having a sale or have coupons for that type of thing first.
2. The games you should get are really based on her personal preferences. Since she likes Mario, I recommend Super Mario 64 DS, New Super Mario Bros., Mario Kart DS, and/or possibly Super Princess Peach. Games have a varying price, normally around $20-35, so again, check your local stores for possible discounts.
3. I personally don't have any accessories for my DS, it has a regular headphone jack so you can just use whichever headphones you already have. Oh, there's the Nintendo USB Wi-Fi Connector, which you can plug into your computer and use to play some of the DS games online with people from around the world, in the same sense as Xbox Live. If you're interested in that, I'm sure someone here will explain it in further detail for you.
4. It has a built-in battery pack, and comes with a charging cord, so you don't need to worry about batteries. Once the battery in it dies, plug it into a wall socket and let it charge for an hour or so.
I'm sure there'll be plenty systems for you when you go shopping, seeing as the demand for the system isn't as high as it was when the system was released. You shouldn't have to worry about the store running out of stock, although they may be out of a certain color, in which case you might have to try a different store. Hope this helped