Does having a bedtime for my child a good idea for preventing a cold?

No. I am personally not aware of any research in this direction. If we believe colds are caused by viruses, i don't see how bedtime routine can prevent that.
No relationship. While having adequate sleep is a good way to keep the immune system strong, i don't see how defining a specific bed time has anything to do with adequacy of sleep /day or the frequency of exposure to & infection by respiratory viruses. Most infants will drop off & doze periodically to make up for any sleep deficit. I do like predictable sleep rituals every night.