If I attack but do not shoot down a UFO, and then it lands and I send in a skyranger, will I be facing less live aliens than if it had landed fully unmolested? Does damaging the UFO have any effect, or must you fully shoot it down?
No, it doesn't.
As usual, the UFOpaedia has all the relevant information, in this case in the article about battlescape map generation.
Specifically, down at the bottom of the page is the description of how the game determines when and how to damage an alien craft:
Landed UFOs have this check not done thus don't get damaged. Without those explosions all spawned aliens are still alive when the battle starts.