To be fair, they are shapeshifted when this occurs so, that's an altogether different argument and is irrelevant to the current question.
That really only solves the how part - the rest are structural inconsistencies which will each try to assert themselves during development, probably with disastrous results. Unless you're claiming polymorph alters your DNA - in which case there are no half-dragons, because the dragon was human at the time.
The more important thing is how the fetus and such would develop which would have to depend on the mother of the child. However, since a dragon is magic, mixing it's DNA can do many different things. Who knows exactly why, it's magic!!
This is also in response to Dan2:
The thing is, magic only does what it explicitly says it does. For everything else we need to assume real world mechanisms and explanations. I mean, I'm pretty sure we'd reject the notion that the stork brings babies in the D+D world barring text which explicitly says "a stork brings babies". (Admittedly, that would solve the half-dragon problem... although it makes one wonder where the stork gets the babies from).
I mean, if we're free to assume anything, then humans may very well lay eggs. Its already common knowledge that the rules never define any negative effects for the "dead" condition, so its clear the writers expect us to apply some real-world logic and intuition. In fact, if nothing works like in reality then it becomes impossible to play, because the players don't have any rational expectation for anything.
I mean, we could assume that there is no gravity, things just 'stick' to the world. And shooting an arrow doesn't apply force on the shaft, rather the bow user casts a magic spell by drawing and releasing such that the arrow travels towards the target. We could imagine a whole lot of crazy, except then we have the problem of exceptionalism - with no general rules, no one actually knows what happens if you try to lift something with a lever. This is the problem with
magic realism as a general mechanism, because only the author/DM understands how anything functions at all. While it can work for literature and cinema, its really awful for a cooperative game.
The writer never says that 'magic occurs'. There's no rules or fluff which explains how gestation of half-dragons works (afaik - not an expert on the Draconomicon). The writer just says they 'mate' and leaves it at that. Which leaves us to fill in the details. Now, either we assume real world processes apply unless something intervenes, in which case all half-dragon fetuses abort. (And we get half-dragons from when there's an intervention - quite possibly handled by the dragon itself) Or we assume that real world biology never applies to anything at all, and whenever the topic of reproduction comes up we have lots of unanswered questions.
And AFAIK, dragon-dragon reproduction is perfectly mundane and works normally. We know it results in eggs that hatch baby dragons. What we know of dragon mating rituals isn't actually all that different from some birds. No (professional) D+D writer, to my knowledge, has bothered to describe dragon sex, but given their equipment and relatives we can make some reasonable deductions. If all this is perfectly mundane, why should we invoke exceptionalism in a dragon-human pairing?