Hey, slow with the math!
What we have to consider is that a human has 23 pairs of chromosomes - all of them code for him being human, and the last pair, the gonosomes (as opposed to the rest, the autosomes), code for gender, amongst other things.
Now, all of a human's chromosomes are likely to be human.
If a dragon had the same number of chromosomes, then a half-dragon childe would have one fully human set and one draconic set.
This may have the following consequences:
If the human set is dominant and the dragon set recessive, then the human set is expressed, the child human, but with the potential to sire/give birth to a very weird child when mating with another such crossbreed. Thus, the first generation is inapparent.
If the sets are co-dominant, or intermediarily dominant, either the attributes of both species are fully expressed (a were-dragon?), or in the case of intermediary gene expression, a half-breed with attributes half way between the human and dragon heritage comes into being.
Likewise, if the dragon set is dominant, then the human genes remain dormant until at least in one chromosome pair, two human chromosomes meet.
Furthermore, parts of the genome can be dominant, while other parts might be recessive. Thus, a crossbreed might always express certain attributes of the dragon parent, say, if the gene set for 'wings' is dominant, then the offspring, whether having one or two of the dragon chromosomes in this locus will have wings. If two heterozygotes (bearing only one wing-gene) mate, they might spawn a child that does not bear wings (25% of the time) as it has inherited two 'no wing' human genes.
The matter is even more complicated in the second generation:
Due to the so-called crossing-over phenomenon, dragon and human chromosomes can exchange segments, thus producing mixed chromosomes with parts of the DNA of both parents.
Now, a spermatozoon, or ovum, has only one set of chromosomes. In the meiosis step when they come into being, they are allocated one of the two chromosomes. Thus, the reproductory cell of a half-dragon parent may get anywhere from 0 to 23 dragon chromosomes. He could spawn a purebreed human with a human mate, a half-dragon with the same, a pureblood dragon with a dragon (or another halfbreed!) or simply anything in between.
Now, this brief treatise on genetics is sure difficult to stomach, or implement in a game, and fantasy uses to stretch realism.
My suggestion is the following
Dragons, the beasts that are an embodiment of magic, should be dependent on chromosomes?
What if their 'cells' or simply 'seed' housed magical patterns, small vortices or sigils, runes of power in microscopic form, that, through a symphony of concordant magic, allow the existance of a being that denies the laws of physics and sometimes even the will of deities.
A dragon, when shapeshifted, still carries the same patterns, they just adjusted to give him another form (so this is no shapeshiftinjg spell, but an ability that must be learnt). When he mates with, say, a human, those small charges find an ovum, and enter, nudging it to life. A half-dragon kid would have a single set of human chromosomes and one of those tiny runes.
The human set would give him the human guise, while the draconic would shine through, and manifest in things like growing wings... a slow metamorphosis. One that can be hastened throughj training, aided by talent and strong will, or learnt through meticulous study.
If such a half-breed mates, he can pass on few or many of the magic chromosomes to his kids.
Thus, they either will have the ability to train a power, or not.
A child of a dragon and a half-breed, or of two halfbreeds, will have a wide array of possibilities of the layout of his pattern. He might have dragon chromosome pairs in a few loci, while the other pairs might be totally human - thus being truly draconic in a few attributes, and lacking dragon power in others totally. He might have a mixed set in say 3/4ths of the chromosome pairs, and pure dragon heritage in the rest, making him more draconic in certain areas than a first generation half-breed.
One whose blood is sufficiently thinned might have one or two charges, thus giving him the potential to learn a few draconic abilities, with the rest forever out of his reach.
Let us determine 23 dragon abilities, and assign each to a chromosome.
A character (half-breed) can either be assigned, or 'purchase' dragon chromosome points, and allocate them.
So, one player might spread 23 points out evenly, creating a first generation half-breed, with potential in all areas.
His abilities will surface over time and with training or need, and be variably developed.
Another might omit certain abilities totally, and spend the points to buy fully draconic chromosome pairs, thus giving him full power from the start in certain areas, while closing his path in others. Such a being might have the strength and fire of the dragon, bnut lack his wings, and magical prowess.
now, another thing I'd add is the Awareness factor: ranging from say 1 to 6, it determines how well the ability is surfaced. 1 stands for potential that is totally unused, thus the being would be totally human in this regard, but with the door to improvement open, while the rating of 4 stands for full dragon power, and 6 for maximum dragon prowess in the given field.
One PC might be strong, armored in scales and have terrible claws, with great prowess in all those fields, while another might have only inherited the dragon's magic talent, well-fledged from early childhood on, and NO visible dragon traits.
Another thought: One who truly masters an aspect of the dragon nature might be abel to achieve Ascension in that field: convert his human chromosome of the pair to draconic, from matter to energy. It might even be possible to convert a chromosome of a completely human pair, though extremely difficult.
Dragons might be dependant on mating with other species, who in turn have to mature into a dragon.
Some cults might consider them a disease, or curse. Some might think them spawn of the devil planted ionto the loins of innocent females.
Some might think them gods.
And they themselves? Rivality, seeking of the self, madness for power, despair at the unwanted change...
The possibilities are innumerable.