My question is.. when indexing the principles names if they used a different name when entering the US and and any aliases they used, do we index it with there name separated with an "OR"?
Yes. From the Given Name Field Help (purple question mark):
Include all aliases, nicknames, or variant names that were given for a person, separating them with the word Or.
Your names will be the following:
GN: Oliver Carl Or Karl Olavi
SN: Kallio Or Hatinen
This instruction is also located under Project Instructions/General Indexing Guidelines/Alias Names.