If electrons sometimes time traveled on a human time scale we could exploit that, eg. by building a huge assembly of electron guns, phosphors and photomultipliers. We could gain knowledge of the future signal sent to the electron guns by watching when the phosphors lit up while the electron guns were switched off.
Depends on scale. Suppose they are rare and show up in random locations with random velocity's random amounts of time in the past etc. As in 1 electron in the observable universe per day.
If all their properties are randomized then I don't see how they can be said to "travel". That would be indistinguishable from electrons spontaneously appearing and unrelated electrons spontaneously disappearing.
This is one of the major reasons I said time travel without information transfer is not interesting. Even if it actually happens, it's not distinguishable from something else.
PS: Quantum entanglement is also less interesting than you might think for similar reasons.
In the case of entanglement though it is actually distinguishable from other, less "interesting" phenomena though, right? That's what the loophole-free Bell tests are about.
