What color is the light on the brick when it is plugged into power and when the xbox is plugged into it? You may have a bad power brick.

It sounds like the box overheated. There is a troubleshooting sequence here: http://support.xbox.com/en-US/xbox-360/console-power/xbox-360-no-power-solution , but it looks like that's going to simply tell you that your xbox needs service. Is your console still under warranty? If so, I would recommend sending it back to have it fixed. If not, try re-flowing the motherboard and putting new thermal paste under the heat sinks.

Have you replaced anything on the xbox, or done any other work to it? If you replaced the original DVD drive with another one that is not "married" to the motherboard, it will not play game discs. The motherboard and DVD drive share a key, it is different for every system. If the DVD drive was replaced, here's a write-up that gives you some options on how to put the original key on your new DVD drive so that you can play games on it again. http://hardandsoftgaming.wordpress.com/2...-a-faulty-xbox-360-dvd-drive-firmware-flash-or-circuit-board-transplant/

Pop out two plugs inside the tailgate opening and remove the two screws under them, this will allow you to "pop" the taillight assembly loose from the vehicle. Once that is done, you can disconnect the electrical plug to remove it from the vehicle. If you are just replacing the bulbs, you don't have to do that, but it makes it easier. Twist the bulbs you need to replace counter-clockwise to remove them, or move all of the bulbs to the new part if you are replacing the entire assembly. Installation is the reverse of removal, the link below has a video that shows the process in good detail. http://www.youtube.com/watch?v=D_LMd9CD7...

Did you get it wet or spill something on it? If the right button is now doing the function of the A button, it sounds like something shorted inside the controller. You could take it apart and see if it is corroded inside, or if it is wet. If it is, clean it up and let it dry out before you put it back together and see if that fixes it.

Does it work properly with the original cable? If so, there may be something wrong with the universal cable. If not, the solder joints on the GPU may need reflowing, here's a DIY on how to do that. Reflowing Xbox 360 Motherboard

Try cleaning the copper battery contacts...it sounds like the controller is not getting voltage from the battery.

It's unlikely that the overheating could be caused by any of the screws being too tight. Usually the two red lights errors are caused by the CPU overheating, the GPU overheater, or cold solder joints on the memory. In your case, it sounds like everything was overheating. Check and make sure your fans are working...if they are not, it will overheat quickly. After you verify that your fans are working, you can try re-flowing the motherboard to see if you can get it working again. Here's a good DIY on how to do that: Reflowing Xbox 360 Motherboard .

Code 0306 means that cylinder # 6 is missing...since that one is drenched in oil, that is probably the problem. Most likely cause for the oil is a leaky valve cover gasket. I've heard that a few people had issues with the valve cover itself causing the leak, so check it closely when you pull it to replace the gasket.

If it is leaking out of the back of the engine, it is probably the freeze-out plugs as mayer stated...low coolant will cause the heater to work intermittently or not at all. Check closer before you get the heater core replaced.