Can you built a settlement in the middle of someone else's road, or vice versa?
Yes. You can build the settlement in the middle of an opponent's road (as long as your own road connects to it and there is at least one gap before any other existing settlements).

Yes. It does affect the longest road: the road stops at the settlement for counting purposes, and starts again at the other side.

Yes. Both players who possess roads into an intersection have the option of building the third road out, regardless of whether a settlement is there or whose it is.