An empty house is a house but not a home. A welcoming apartment is a home but not a house.
"A House Is Not A Home" is the title of an autobiography of an ex-prostitute. When she was a prostitute, she lived in a hous but it was not a home.
Fun fact : In french, we don't have a word for "home" so when we want to say "I go home", we can either say "I go to the house" (je vais à la maison) or, badly translated, "I go to my place" (je vais chez moi). But "at home" is a concept that French lacks of, I think.
They're spelling variations of the same word. If you're going for a moderate/conservative Bokmål, the obvious choice is "hjem".
So would that mean that "Heimdall" is..."house guardian" or something similar?
"Heim" could mean "home" or "world" in Old Norse as well, but the meaning of the name "Heimdall" isn't quite clear. One possible interpretation:
Navnet Heimdall betyr kanskje «den klartlysende» eller «den som lyser over verden». Dette passer godt med kallenavnet "den hvite ås", og at han omtales som «den hviteste av alle æsene».