Real photons travel at the speed we call "c"≈299,792,458 m/s in a vacuum, and can last indefinitely without decay (traveling millions of light-years, in astronomy).
Virtual photons are used to explain some atomic reactions, where no real photon is a product of the reaction. They only exist for a very short period of time and travel a very short distance; these are temporary deviations from the familiar macroscopic physics permitted by
Heisenberg's uncertainty principle.
It is possible for a real photon, travelling at c, to interact with a real electron in an atom. Because the photon could be considered as propagating as a wave, its precise position is unknown; the wave function represents the probability of finding the photon at a particular position.
One of those positions could be at the position of the electron which absorbs the photon (bearing in mind that the position of the electron in an atom is not known precisely, either).
So I suggest that a photon does not
need to exceed c (
or travel backwards in time) in order to interact with another particle.