Do other planets have days and nights?
Yes, any planet that is orbiting a star can be said to have days and nights. Whichever side of the planet is facing towards the star is daytime for that planet; the other side is night.
All planets rotate about their own axis as they orbit their parent star. Most rotate in such a way that the side facing the star is constantly changing. However long it takes to rotate once is one day for that planet. Note that there is a difference between the time it takes to rotate relative to the star and the time it takes to rotate relative to the rest of the Universe. For most purposes we would say the time it takes for a planet to spin around and be facing back towards the stars is one day (i.e. "solar day"), the less-often definition is called a "sidereal day".
Some planets are "tidally locked", which means that the same side is always facing towards the parent star. In this case it's always daytime on one side and always night on the other. These planets don't have solar days but they still have sidereal days.
Rogue planets (that do not orbit a star) are the only planets that don't really have days. You could say that for these planets it's always night.