Qestion 2 is interesting as an example of how one could go into more and more detail into the phenomenon and get different answers for each "level" of detail. This is particularly true here because we ask about the force at the very instant when the block is about to move upward. In practice, there would be no "instant" but there would be a transient process where the force would vary in interesting ways before settling down into ##W-\rhoVg##
If we apply an increasing upward force on the rope, a moment would come when water would try to rush into the gap between the block and the floor. You could think of it as a transient partial vacuum under the block, or as a bernoulli pressure drop that results in a net downward component.
If the slack on the rope is taken up infinitely slowly, then the bernoulli / vacuum effect would always be negligible and the tension would climb smoothly from zero to the archimedes value.
