Question: We have been introduced to 5 characters (all Impel Down Jailers) with "Awakened" Zoan Devil Fruits. Minotaurus is the most famous of them. A normal Zoan fruit allows the user to transform between Natural (normally human), Animal (based on the devil fruit type), and Hybrid forms. I can't, however, remember any of the "Mino" guards in any form besides a exageratedly strong version of the Hybrid one.

At the time I thought "Awakened" meant that the fruit was always active so they were stuck as partially animals (as well as give the strengths Crocodile mentioned). With Doflamingo's Paramecia fruit, however, awakening appears to just unlock new abilities. This made me doubt my previous understanding. Wiki doesn't help and I don't have time to rewatch everything.

Do we know whether Awakened Zoan Devil Fruit users retain the ability to transform into multiple forms?

If the answer to this is "Oda hasn't told us yet", that is sufficient.

Answer: 
My theory about its "real name" is, either Towako, the name of hakumen's avatar, maybe Hakumen named its avatar the real name for self-satisfaction? and the second probable name, is kirio/Inasa, as in Kirio Inasa, the boy whom it kidnapped. after the kidnapping, it probably change his name as it likes. 

I personally think, Towako is the "Real name".
In my opinion i think Hakumen’s real name might have something to do with the baby shown as he vaporizes and yin/yang because in the end he does not fall as negative ki which creates other youkai, but he floats upwards as positive ki. Maybe in the end he died a peaceful death and he could now be reborn as a human, not a mass of hate.We can also assume he would have a woman name since his voice when he whispers “The name i was called was...” is a feminine one.