i think opening the housing and leave it that way is not a good option since the air-ventilation inside the housing isn't given any more, due to there's no housing. what you can do, go to a pc-shop and try to get a better fan and install it to your xbox, or just place a normal room fan in front of your xbox to do some passive cooling. do you have your xbox standing in a shelf or is it in an open spot, is it confronted with direct sunlight, etc. if so try placeing your xbox to another point in your room...
you could also try to install a watercooling system, but i guess that would be as expensive as a new xbox (the small one) which has a better cooling system and rrod is practicly impossible
edit:
on the connection error thing, it could be that your xbox suffers of a cold soldering joint which means once its hot the connection between the pcb and the component for the network is broken... while when cold connection is given