Can you build a settlement on someone elses road?
No. If you build a new road, it must always connect to one of your own roads, settlements, or cities.
Can you build a settlement in between two opposing roads?
Absolutely correct. The presence of an other player's road does not prevent you from build a settlement there...
Can you break someone's road in Catan?
Yes, settlements do break up other player's roads. If the blue player has a road that is six segments long, and the red player builds a settlement between two of the blue roads, than we say that the blue player's road is "broken". A broken road can not be counted for the Longest Road card.
Can you interrupt someone's road Catan?
You are correct. You can break an opponent's road by building a settlement on an open intersection along his road!