Do all the D' girls have dragon "mates"? Some of Yuu's classmates are shown to have dragon "partners"/"suitors" (to avoid a more heavy term), and the plot runs around avoiding having the dragon meet the designated girl.

I am talking about when a dragon comes for a girl and her Ryuumon glows.

Examples are Iris and Leviathan, Tear and Basilisk, the girl Mitsuki killed (she turned into a dragon in the backstory).

But the school has dozens of D girls. Are there that much dragon beasts rampaging around?
Probably not, as I cannot imagine that Yggdrasil is in the same realm as mankind, I would imagine there are multiple or perhaps even an infinite amount of realms. That would also explain the sudden appearance of the Dragons.

Unfortunately I have no real evidence of this being the case. My comment is based on mere assumptions after watching the aired episodes of the anime series.