Hmmm...not sure on that. The rule says, "The batter-runner is awarded first base and runners advance one base if forced." If the INT removes the force for the runners ahead of her, then the run would not score. Hopefully, we can get a clarification on that one.
A run cant score, yes, the batter/runner is awarded 1st base but all other runners must return to the last base occupied at the time of the interference unless forced. No matter what runner was hit by the batted ball, the runner at 3rd is either out for the interference, or returns to 3rd base because of interference by another runner. The only possible scenario where a runner could score was if they happened to cross the plate before the other runner was hit by the batted ball for the interference call and that would be one extremely fast runner, like Flash Gordon quick.