Applepie, the fan is electrically driven and controlled by a heat sensor in the water system,even after a short drive in warm weather the water will get to normal working temp causing the sensor to tell the fan to come on,the fan will stay running,even when the engine is stopped until the water is cool enough for the sensor to switch off the fan,how long does it run after the engine is off,bet its only a couple of mins??
As for your exhaust,welding is a good fix if the break is in a part of the pipe where the metal is thick ( most of the pipe is thin,especially if its old and rusty---they tend to rust from the inside out!)
May be better to get a exhaust company to look at,like kwik fit they would probably do you a better deal than an ordinary garage.