it is because of heat
Yes, Xbox 360 will still have Live because its part of the Microsoft Company and part of the Xbox company also
yes it can over heat after a period of time
Cables? Dude, you only need an Ethernet cable to connect to XBOX LIVE, connect the Ethernet cable to the right most back part of your 360 (assuming your 360 is on its side, if it isn't its the top part) and plug the other end into your router/modern and then connect XBOX LIVE.
It has to do with their Xbox Live services, all of Microsoft Xbox's status can be found in the related links below. For the most part, they are aware of it being down or having difficulties, it's being worked on.
about 900
xbox 360 elite
The elite has 6x the memory and doesn't over heat.
Any console will overheat, it depends what you are doing on it and where its placed. If the Xbox is in a small area like a enclosed cabinet, you will find that the area will get warm and will increase the heat of the Xbox. However, the new Xbox 360 Slim has a better cooling system, which seems to help considerably.
You can modify the hard drive size. The hard drive is located at the bottom of the Xbox. There is no other part to modify the Xbox a part from taking the console to bits.
if your talking about remote that controls the media part of the xbox 360 then yes, but if its the xbox 360 controller then you would have to sync up the controller with the other xbox
most games you can play but mostly the new games also you should have a gold Xbox live membership and when you do make a account and sign in to Xbox then it should update your Xbox which should let you play most Xbox games on Xbox 360