A star with no nuclear fuel left would give off light according to its temperature (same as a planet does).
Basically, everything gives off a range of light according to its temperature (google "black body radiation").
eg red stars are cooler than blue stars.
A light bulb gives off light because it is hot.
, some things give off light at certain exact frequencies
because their molecules are vibrating (eg sodium lamps).
No, being hit by light does
give you both energy and momentum.