Starcraft – What are all the different *titles* for finishing the campaign

Once you've completed Lost Viking, are you able to select higher difficulties without first re-completing the game on the easiest setting?

There are no difficulty settings that you can select. The game gives you the option of starting back at the first level and gets progressively harder each time you choose to loop through it. Enemies get a little harder to kill and a little faster, and certain in-game elements change. For example, the 3rd level has laser barriers you have to dodge on subsequent play-throughs (level6, 9, etc).

Presumably there is a score multiplier when playing the harder modes; what are they and how many difficulties are there?

Nope. Enemies are worth the same number of points regardless of how many times you've looped through the game.

When returning to Lost Viking, my "high score" is always 0. Is this retained if you save after leaving the arcade game?

It's always retained my highest score from within that campaign. I haven't checked to see if your high score persists to game sessions in other campaign play-throughs.

Once you beat the campaign, all you can access are the mission archives. If you never completed a mission in the first place (and thus can replay it in the archive), the only way to access the rest of the missions is over a new campaign.

Alternatively, if you have a pre-Char save of your campaign, you can load that, and go play the missions you missed (before you head off to Char, this time!)