In Regicide, how do you become the king

without one preventable Marine death

Try the mission on easy, you should be able to save everyone. Except those that die because they are scripted to.
Those scripted to die will always do so under the excact same circumstances, even on higher difficulty.
Also, here's a video guide: Link

From personal experience, the difficulty scales in a small way based on the number of players, but it's nothing compared to the benefit from having additional players.

Co-Optimus' FAQ for this game backs me up:

Do enemies scale based on the number of players?

There seems to be some limited scaling in the campaign. Spartan-Ops also appears to scale slightly based on the number of players. You'll want to change the difficulty though for maximum effect.

It's hard to quantify exactly how the difficulty scales, but the developers of Halo: Reach implemented this system:

In Reach, Jarrard said, enemies will toughen if they face more players, exhibiting not just more damage resistance but getting smarter, more creatively using cover and generally being trickier foes.

Halo 4 re-uses parts of the Reach engine so it's likely that this information still applies.

I didn't notice an extreme difference in damage resistance going from solo to co-op. It's also relatively hard to gauge the "intelligence" of the AI. They seemed to use cover, flanking, and dodge/teleport just fine in single player, especially on Legendary.

On the other hand, the benefits of co-op are many, and easy to determine. Having multiple weapons, and giving the enemies multiple targets, for instance.

Another prime example of how much easier it is - as long as you don't turn the "Iron" skull on, and as long as one player remains alive, others will respawn on them (or nearby) if they die. In single player, (or in co-op with Iron on) you'd have to go back to the last checkpoint and start again. This can make even difficult engagements fairly easy, as long as you play conservatively. (I believe this is also why there's a separate achievement for solo'ing the campaign on Legendary)