Is the performance of a CPU affected as it ages?
Even if we periodically cleared away all the dust in the CPU processor?
And if so, what is the rate of slow-down? Would we be able to see it in task manager?
No, the CPU runs at a fixed clock speed as determined by the BIOS and/or motherboard configuration. A build-up of dust on the CPU may cause it to overheat, which can eventually result in it locking up or failing altogether, but there will not be any gradual reduction in clock speed.
It is theoretically possible to have a CPU based on asynchronous logic with no fixed clock which could run as fast as the temperature would allow, but they are not used in any consumer machine that I am aware of.