Should Git track the "typings" folder? (typings.json)

You should think of typings folder as node_modules folder for TypeScript, therefore it shouldn't be checked in. If you want to simplify your build, add

{
   "scripts" : {
      "postinstall": "typings install"
   }
}

to your package.json file then typings install will run automatically when your run npm install. Don't forget to check in your typings.json though.

Since it is generated content I believe we should not track it, but I'd like to have some feedback about it.

I always check it in. Until it's a part of npm, that is what I do. Also helps me overcome bad definition files and chug along with my project. (e.g. Change type imported from typings definition)