Gothmogs and dragons and noobs, oh my

[The Might]

Indeed, he corrupted an already existing race of dragon-like creatures.
I could personally imagine this corruption as similar to what Sauron did with the winged beasts during the Third Age:

"A creature of an older world maybe it was, whose kind, lingering in forgotten mountains cold beneath the Moon, outstayed their day, and in a hideous eyrie bred this last untimely brood, apt to evil. And the Dark Lord took it and nursed it with fell meats, until it grew beyond the measure of all other things that fly; and he gave it to his servant to be his steed." ~LOTR

Perhaps this pre-existing race Melkor used to devise dragons wasn't in any way as strong or powerful as the dragons themselves, but by using his power he might have been able to "update" them.

However, as far as details on this race are concerned I can only speculate. For example in the Hobbit, Bilbo makes this affirmation:

"Tell me what you want done, and I will try it, if I have to walk from here to the East of East and fight the wild Were-worms in the Desert"

Now, considering the reputation of Hobbits as far as such things are concerned, I tend to doubt that Bilbo actually knew what he was talking about, but there might be some hidden truth here as well.
The Last Desert was thought as a dry, desert region in the Far East of Middle-earth. To me it seems a likely possibility that Melkor used such a race to devise the dragons. But as I said, I am not implying that this theory is right, just trying to give an example of a possible race.

And now to the Gothmog question.
I very much doubt he was an Orc, since as already said there clearly was a big difference in status and authority between him and the other Orcs, as already mentioned above, leading me to believe he was of another race.

But I also very much doubt he was a Nazgul. The namelessness of the Nazgul is one of the key characteristics of the Nine. The fact that their true identities and names were forgotten seems pretty obvious. (except the strange example of Khamul, and I still can not fully understand why Tolkien chose to name him)

This leaves me with the most obvious choice - he was also a Black Numenorean. If, as already said, they were capable of reaching such high ranks and gain so much power, I see no reason why Gothmog wouldn't have been one as well.