Protective gear....to visit a sick patient...possible contagious disease...no mask?
I would think that the alchemist’s garb is magical so it protects the whole body, including the head and any other exposed areas. Elliot did say that alchemists have to wear protective gear since they deal with [toxic] chemicals for their research. Noxious fumes from chemicals can be inhaled, so hence why I think the cloak is magical.
So the emperor wanted our MC to marry the prince because he knew she has magic, and the prince needs magic to ascend to the throne.
Idc about the prince, I'm wanting to see the little elf kid be adopted. He is so cute. Please let him meet the girls and they play together