Find a graph with five vertices and with exactly 22 cycles?

Check your definition for "cycle." I think no repeats are allowed, so your lower figure would have only 2 cycles, right?

I'm confused about one thing, and that's the order of the path. I thought K3 had only 1 cycle, but http://oeis.org/A119913 says it has 2. If the vertices are A, B, and C, apparently it counts ABCA and ACBA as different paths, even though they all have the same edges: AB, BC, CA. When is this counted as 1 path, and when two? And which way of counting are you going by?

I really like graph theory questions, though I've never had a course in it. But the terminology ambiguity drives me a bit crazy.

For a 5 vertex graph with 22 cycles, I'd start with K5 and count the cycles. Then get rid of one edge and see how many you still have. Then another one, until you get down to 22. I think that would be a way to solve it, and a good learning experience.