Yes that would be a big problem and will cause freezing and rebooting, also could cause damage to any part of you computer when it decides to blow cause it is overheating.My 350Watt PSU fan appears not to be operating - would this be the cause of the excess heat?
Dunno, would need to know exactly what you have, how about posting that Everest scan, copy/paste it here.I have an old 300 watt PSU, which I can dig out and if it's working, could I fit this to the PC.
I'm concerned about going down in wattage as I've got a new gfx card and extra optical drive waiting to be fitted. Will the 300 PSU be sufficient?
Read the label on both PSU and type that info out here as well.
Do that then we can help you.