1. No, you gain 8. The creature will do it's full power in damage unless something like protection stops it. If it had trample, it could only *assign* one damage to the creature and the rest would carry over to you (and you'd still gain 8).
2. It doesn't matter. The card will do as much as it can possibly do. He'll have to discard as much as possible. Cruel Ultimatum is a very straightforward card like that, but certain others, like Cryptic Command, are much more complex. For instance, if you choose two modes of "counter target spell" and "draw a card", and then your opponent Remands the spell, then Cryptic will no longer have a valid target and become countered. It won't resolve, so you won't draw a card. However, if you chose "counter target spell" and "return target permanent to its owners hand", and your opponent still Remands, then that portion of the spell will become invalid. However, since Cryptic still has a valid target, it will try to do as much as possible and the permanent will still bounce. It seems weird, but the thing to remember is that a spell with no valid targets becomes countered. The draw effect is not targeted, so if it loses its only one it won't happen.
In both the cases you mention, it sounds like someone trying to be sketchy about the rules. It would take a very skewed reading for a player to misinterpret them like that.