Settlers of Catan for 2. Rolling the dice twice.
I would say no, you cannot.

The rules explicitly state what you can do after each roll:


  Immediately after each of the two dice rolls, the two real players obtain resources and/or move the robber (a “7” result).


Since there is no mention of building or trading, I would say you could only build and trade after finishing all rolling.

That would make the order is Roll > Get resources > Roll a second time (repeating as necessary to get a different value than the first) > Get resources from second roll > Continue play as normal.

Think of it this way -- you are player two, and the other human player is player four.  You are effectively rolling for player one's roll, then you roll for yourself and can buy and trade as normal.  Then the other human player rolls for player three, and then they roll for themselves and can buy and trade as normal.