The hard drive light shouldn't be on all the time, but if everything else is working normally, I think it is quite possible that you have a connector wrong, it's easy to do with the front panel connectors. Is the power led on permanently as well? Perhaps you have got them swapped over.
The led being on permanently isn't a problem in itself, only if it is indicative of another fault.
I have never heard of an "expired" cpu causing the hard drive led to be permanently on, unless there is something else wrong at the same time. Usually a dead cpu results in the computer not even booting.
I am suspicious of those temperature figures. What are you using to measure them? The cpu usually runs about 10° higher than the motherboard, not the other way round. A cpu temperature of 50° to 60° is normal, and after a time of running the motherboard temperature would normally be about 10° higher than the ambient air temperature.