Haku was a computer hacker, and his older brother is the one who would later become the emperor. Haku was the one who hacked into the other facility and stole data about Iceman Project (that would say the research on Hakuoro).
Anju is the emperor's daughter and a clone of Haku's niece. She the next to become the emperor after the emperor's death, but some wants her dead and other use her as a puppet. At the end of Itsuwari, she has been poisoned so she is bedridden and can't talk, so Haku took her to the country where Oshutoru comes from to keep her safe.
No, it looks like everyone have turned into slimes, but it looks like Haku and the Emporer escaped it due to drinking some medicine that the emperor had created before Haku went into cold sleep. Emperor's wife and daughter turned into slimes. Even cold sleep can't prevent it, since ones they wake up they turn into slimes.
Haku were supposed to sleep for a shorter while than he did, but people started to turn into slimes.
This is how the emporer's wife and daughter looked like. He would later clone them, Anju being the daughter and Honoka being his wife.