1. There’s a sort of diffusion process going on. Photons from the core have some mean free path as a function of radial position (and, obnoxiously, of wavelength as well, so maybe we ignore that). You could calculate the mean time for a hypothetical object emitted from the core and traveling according to those mean free paths to escape.
2. You could imagine you have marked a photon and watched it travel. This is quite problematic. First, photons in thermal equilibrium obey Bose-Einstein statistics because they are indistinguishable bosons, and anything that could mark them would change the statistics to that of distinguishable particles. But whatever, the temperature is high and maybe this doesn’t matter. Also never mind that those core photons are mostly much shorter wavelength than the photons we see. But you can still imagine. (The answer is probably quite similar to #1 since this is sort of the same problem depending on how you think about the interactions with matter in the sun.)
3. You could calculate how long it would take to notice anything if the core suddenly stopped fusing.
