Do photons moving at $c$ slow down when entering a medium or take a longer path and only appear to slow down and what happens to time?

Question

Imagine a photon leaving a vacuum and entering a medium, say, air. I have 2 questions:

1. Some claim that the photon is slowed by the medium so its speed becomes less than $c$. Is that true or does refraction cause the photon to take a longer path so that it only appears to slow down?

2. Since time does not exist relative to the photon when it was traveling at $c$ in the vacuum, what happens to time relative to the photon when it enters the medium?