As far as I'm concerned, Half-Dragon = Dragonborn, and will be so when I run HofDQ. I'm not going to bother differentiating, other than for narrative effect. That said, our game doesn't have dragonborn as a playable race, so we won't have the necessity of differentiating while playing.
Half dragons are the offspring of one dragon parent and the other parent of another race.
Dragonborn were created in the distant past and are their own unique race now.
The answer to this one is setting-dependent. IIRC, in some they were deliberately created by Bahamut, in others they were more an accidental creation from the death of Io (i.e.: when Io was slain, dragonborn appeared from the drops of blood that touched the ground).Was there some reason they were created?

(Dungeons & Dragons)
Rulebook featuring "high magic" options, including a host of new spells.