tkotw, that's mostly correct. But, you seem to be implying that II is impossible. (I'm inferring this from your description of III, but NOT II, as "conditional".)
If you are saying that II is impossible, then that's wrong.
The problem there is that you're considering only 3 values. Since there are actually 15 values, there are 7 values that are > to the median, and 7 values that are < to it.
In other words, the 7 biggest values can move up or down to compensate for each other.
E.g., all of them could be $190,000, as you've described yourself.
However, it's also possible that you could take away, say, $50,000 from one of them and add it to one of the others. In that case, one value would become $140,000, another would become $240,000, and the remaining five would still be $190,000 each. Your three-value model can't account for such possibilities, since a single number can't increase and decrease at the same time.
Because of the way this problem is formulated ("must be true"), you ultimately get lucky here, and this doesn't matter. But, if the problem had said "COULD be true", then you'd get the wrong answer if you think II is impossible.