Can a soldier receive multiple promotions in a mission?
This might be anecdotal, but yes, you can:
I ranked all my squad members to Colonel on the downed Avenger mission, by simply refusing to destroy the signal beacon and farming the incoming assault drops by setting up Overwatch traps.
Took me an hour to do so. After returning to the Avenger I was greeted with multiple rank ups after promoting the soldier (meaning went all the way from Squaddie to Colonel)