Q: Do any of the multiplayer Seven Islands rules carry over to the solo game? Do I get to draw extra cards for playing a Ship or Beacon?

A: Only the Basics of Island Building section is common to both versions. If something is in the multiplayer version that’s not explicitly mentioned in the solo version, then it’s not in the solo version.

You don’t get the Icon Bonus for playing Ships or Beacons to the table. Likewise, in the solo game, you can’t start a floating island with just any type of terrain, only a Beacon.