The speed of light does not change for each given reference frame, i.e. it is constant. However, space and time are observed differently depending on a given reference frame.

With regards to your second question, photons experience no time whatsoever. From the photon's perspective, they travel the universe from the point of emission to the point of absorption instantly.

If you want to understand this, I highly suggest watching this video explaining reference frames: https://www.youtube.com/watch?v=aRDOqiqBUQY

And this playlist of the excellent PBS Space Time series: https://www.youtube.com/watch?v=YycAzdtUIko&list=PLsPUh22kYm...