Once you've completed Lost Viking, are you able to select higher difficulties without first re-completing the game on the easiest setting?
No. The game consists of three levels. Once you beat the Terra-tron for the first time you're given the option to replay those same three levels on a harder difficulty setting or to quit. The second and any subsequent playthroughs continue to add to your current score. I've not yet been able to confirm how the game changes.
Presumably there is a score multiplier when playing the harder modes; what are they and how many difficulties are there?
I now doubt the existence of score multipliers and there's no reason to think the game can't theoretically be played indefinitely.
When returning to Lost Viking, my "high score" is always 0. Is this retained if you save after leaving the arcade game?
It would appear the "High Score" is broken: in my experience there are no circumstances where it will reflect your previous best.
As an aside: If you have played Lost Viking and wondered about the significance of the Terra-tron, it formed the basis of one of Blizzard's 2009 April Fools jokes, complete with streaming video (annoyingly, you have follow that link twice if you've not visited their site previously).
More trivia: The Lost Vikings is side-scrolling puzzle/platform video game developed by Blizzard. The Starcraft II minigame is obviously a play on that.