|It's quite likely that temperature could have damaged it.|
Temperature is usually main casue of CPU failure.
Depending on the fault will cause the system to fail differently.
Normally a CPU that reaches the max operating temperature initiates a "controlled" shutdown of the system to prevent long term damage. However, I have seen faulty PCs freeze as you described because the CPU and/or motherboard is damaged to the extent of preventing this safety shutdown.
Being a laptop makes it even tricker to diagnose.
I'd suggest you definately have the fan replaced.
Unless you have a spare laptop CPU it'd be enclined to take it to a reputable repair center and have the CPU tested.
But based on what you've said I think you could be looking at a CPU or motherboard fault.
IT Services Manager