Android - How to close (shutdown/kill) the Android web browser?

Closing the browser has no benefit. The android subsystem manages the memory on the device really well and will free up memory from background applications if it needs it. As I understand it, it does this by saving the application state in a database on disk. If you run the application again, it can then rebuild its state from the info in the database.

If you want to stop the browser from downloading and you can't see the "Stop" button, press the menu button. This will bring up the address bar allowing you to press the stop button. Although if the address bar has disappeared then chances are your phone is not downloading anything, unless the site you are on has Javascript on it which is causing it to refresh elements regularly. If this is the case you are on your own.

You shouldn't want to close applications manually. Just stop loading the current page with the Stop menu item, and use the Home key. This will leave the app running in the background, but without costing power or wasting your download limit.