Does the Run Score?

[tibear]

Situation: R2, R3 with 1 out. High fly ball to F4, both runners are running with no attempt to go back towards their bases. F4 makes catch and R3 crosses home plate and is walking to the dugout when F4 touches second for the third out.

Two questions:
1) I assume the run counts because nobody appealed the runner at third.
2) Do we have an obligation to tell the official scorer that the run scores? If so, when do we tell them so as not to alert the defense to appeal.