Q: Can we construct a complete graph of the 8-puzzle problem space as we can for the FWGC problem or the water jug problem?

Such a graph would have to include every possible state of the 8-puzzle.

Q: About how many states is that?

The number of possible arrangements of tiles is 9! = 362,880.

The number of states is smaller (not every random arrangement of tiles can reach the final state) but still too large to enumerate by hand.