On catcher obstruction, the batter is given the benefit of the doubt - he was going to get a hit.
Even if the batter DOES hit the ball, play continues until you see the result. If all runners advance and BR gets to first, the play stands just as if there was no obstruction by the catcher. If all runners do not advance... say the BR flies out or is thrown out at first then play stops. Now the offensive coach gets his choice of the results or the umpire awards the batter first base and advances other runners only if they are forced.
If the batter DOES NOT hit the ball, play is dead.
I tend to think of a dropped 3rd strike as a hit (given the necessary conditions) - the BR has opportunity to advance. If the BR tries to advance or other runners try to advance, I'm going to let play continue. If the BR stops AND ALL OTHER RUNNERS stop due to the obstruction then I will kill the play and award as appropriate.
In general, I'm going to let the play continue so when it is finished I will have a choice and can clean-up the mess at that time... I can always go back and rule the ball dead at the time of obstruction... if I need to, or feel that is the right answer.
I am not going to feel the play is done if no one has control of the ball (due to the obstruction the ball gets behind the catcher and runners are advancing on the passed ball... play continues and we will figure it out later). I would never call a dead ball unless there is an absolute reason that the ball must be dead and those reasons are specifically spelled out in the rules.
Perhaps the runner from 3rd will score (or be tagged out) and the BR will stand there and not attempt to run or get to 1st base safely. Now I've got a choice for the coach... results of the play or dead ball at time of obstruction and BR gets 1st others go back to base and advance only if forced due to BR being placed on 1st.
Pitch the ball quickly before we change our minds!