#1 is easy. If x & y are in AUB then you have two cases.

If both x & y are in A or in B then there is an arc from x to y. WHY?

If x is in A and y is in B then there is a z in A intersect B. There is an arc from x to z and there is an arc from z to y. The union of those connects x to y with a path.

For #2, I would need to know what theorems about paths(arcs) you have.