Dragons mate. Dragonborn are mules or Hinnys.
Varies by setting. Some settings don't include dragonborn at all. (For example, there are none in the Homeric mythos.) The version of dragonborn which the 5E PHB describes, pages 32-34, are a species and generally live in clans. I imagine that dragonborn PCs are often exceptions, such as exiles or clan members on "detached duty" (just as Legolas and Gimli were sent to act on behalf of their peoples) or studying some skill which can be best mastered by spending time outside of the territory of one's clan.