As Mizuno said, the little nutcracker girl is the doctor's assistant. Her name is Ramia (Lamia?). She doesn't like the small room or how Oga smells (due to falling in the river). Incidentally, he comments that her kick hurt more than Toujou's punches
The doctor is Forukasu (Folcas?) Rachmaninov, the former Imperial Court Doctor. Seems the guy can read minds since he gets pissed when Oga starts thinking how the guy doesn't even really have a design (more 4-th wall breaking). He really hates the human world and so came like this - his real form is apparently "awesomely cool".
Baby-BL has the "King's Fever" (a made-up word). Among the Imperial Family, the stronger their latent demon powers the easier it is to catch. The doc says it's like "chienetsu" which my dictionary says is "(1) teething fever; (2) developmental fever; fever that brings with it an intellectual or psycho-developmental growth spurt". Anyway, the doc says it's a sign of a growing child and Baby-BL has gotten "King's Fever" a number of times before but not in the human world.
In general, because he's here in the human world, he needs a human's help to bring out his demonic powers, but now because of his growth from the fever (which is higher than normal due to him being in the human world) his demonic powers are now too strong for Oga. Basically, Baby-BL unconsciously cut the link between him and Oga to prevent Oga from being killed.
Anyway, it seems there's not necessarily any particular reason Baby-BL ended up with Toujou, even though he's really strong (Hilda watched the fight btw). From the flashback it seems Oga treated Baby-BL like a stray cat (they normally avoid him).
So, in the end, Hilda intends to help Oga grow as a parent (ie become stronger) so that he won't lose again (Oga says he hasn't lost yet). Ramia says then let's get started, and pulls out a pistol...
