Why half of the medians are greater than the median of medians?
Because $x$ is the median of all the $\lfloor n / 5 \rfloor$ medians obtained in Step 2, it is smaller than or equal to half of those medians by definition of median. See the figure below (from CLRS, the 3rd edition; Figure 9.1).

The problem with your example is that you are finding the median-of-medians $x$ recursively, however, without completing all the recursive steps. You should not stop at Step 3 (the "median of medians" step) in the recursion. Continue the remaining steps. In particular, in Step 4, you will use the "Partition" procedure to establish relationships between the medians obtained in Step 2 and this median-of-medians $x$.

For your added problem: You find it recursively. Why is the time complexity $O(n)$? Well, you can follow the analysis in CLRS.