I'm going to start by looking at the first and last brain-twisters. In my opinion, they're kind of the same argument. The contradiction comes from a lack of clarity in the definitions.
If an object is "unmovable," by definition that means that NOTHING can move it. Similarly, if a force is "unstoppable," that means that NOTHING can arrest its motion. Clearly these two things cannot exist in the same universe, so the hypothetical situation doesn't even make sense. It would (almost) be like saying, "What if gravity were both attractive and repulsive?" According to what we know of physics, this can't happen. The question is meaningless.
(For what it's worth, I'm working really hard here not to throw in a discussion of coordinate systems and how motion is always measured relative to something else.)
Similarly, the "omnipotent" being needs a good definition of "omnipotent." What do you mean when you ask the question? Could this omnipotent being increase its own power? In which case the conflict dissolves - the omnipotent being could create a stone it could not (initially) lift, and then increase its power until it could lift the stone. There needs to be a baseline of power against which to measure the "unliftable stone;" but this contradicts the idea of a truly "omnipotent" being. Again, the argument is meaningless without further clarification.
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