05 Mini S problems.
 

So recently, my 2005 mini s (manual) has been just cutting off randomly when in neutral. Also when I cut my car off, my fan keeps running for a few minutes. And then today I was just driving and my engine temperature gauge went all the way up and then went right back down to normal. There are just a lot of weird things happening and I was wondering if anyone has any ideas on what could be happening to it. Thanks! :)

Your fan Running issue could be a problem with low coolant chk coolant level when cold. they have a very common coolant leak @ plastic thermostat housing to cylinder head they leak coolant (very common)
Good luck

The temp gauge showing getting hot then going cooler quickly could be the sign of a sticcky thermostat. An easy fix, but not sure how easy on a mini.