If I purchase two identical CPUs, and use one long term (say one
year), will it be identical in speed to the unused CPU?
Most likely, yes. The speed a CPU runs at is variable, and set by the end user (although usually set automatically as per the manufacturer's specifications). However, you might find that at the end of the first year, the unused CPU (assuming they were truly identical to begin with) overclocks better than the used CPU. This effect can be attributed to transistor aging, which you hinted at later in your question:
While a CPU has no moving parts (other than the external fan), it does
have circuits that can be damaged by heat, and voltage spikes. Lets
say that after a year of intensive use, the circuits degrade and fewer
electrons can pass since the pathway is narrower, etc.
This is exactly the case, and is precisely what happens after a CPU is used.
Similar to a vehicle, there is some wear-and-tear on the conductors as electrons pass through them. Heat also affects the transistor aging, which is why the CPU die is designed for a particular range of operating temperatures. During operation, the electrons have to tunnel through some layers in the semiconductor materials, degrading them over time. This causes the switching speed of the individual transistors to increase over time, making them "slower".
However, as I said before, the CPU speed is set by the end user. It's a synchronous digital circuit, and will run as fast as you tell it to - even if the propogation delay exceeds the switching time, and the computer crashes. This is what will happen as a CPU ages. Over time, the various sub-units in the CPU will take longer and longer to finish their computations, leading to instability in the CPU.
This effect can be mitigated by slowing the clock speed down, making the CPU slower but compensating for the increased propagation delays. This effect can also be mitigated by increasing the CPU voltage (causing a reduced switching time for the transistors, allowing for a higher clock speed), but raising the CPU voltage will only cause the transistors to age faster.
This is why we say a processor gets slower as it ages - the processor becomes unstable at higher speeds, requiring you to lower the clock speed over time. The good news is that this effect is usually noticable on a timescale of years.