I don't recognize the chassis, so can't recommend a specific body. If you have a decent RC hobby shop close by, you could just take it there and get help fitting whatever bodies they have in stock.

The 27mhz, especially the crystal radio sets are quite old now, and may be hard to find or unreliable at best. You should be able to get any 2-3 channel car radio set and just replace the receiver you have with the one that will be included in the box. This one is priced very well, and should be an excellent radio for $37.

For the battery, that charger will work, the connector is a Standard / Tamiya plug. If that's the battery that's been in the attic, it's probably toast though, so again, off to the hobby store. You'll want a 7.2v NiCd or NiMh, unless you want to upgrade to LiPo, which will require a different charger. I'd stick with NiCd or NiMh for a kid though, as they're less hassle with charging safely and discharging to store. Here's One for $18 and looks to have the right plug. A battery with higher mAh will last longer running the car. This Charger will work fine, more expensive chargers will have more options, usually be able to charge faster, or different battery styles. I'd buy 2 batteries, run one and charge one, or charge both and go to the park.

Good luck and have fun!