Double Play on Tag Up?

 
With runners on third base (R3) and second base (R2) and one out, the batter hits a line-drive into left-center field. It looks like the ball with go to the fence for at least a double, so both R3 and R2 begin to advance. However, the centerfielder makes a diving catch of the line-drive, and then he gets up and throws to second base to easily double-up R2 to end the inning.
 
R3 touched home before R2 was doubled-up at second base. Does R3's run count since R3 did not even attempt to properly tag-up at third base before advancing to home?
 
Call
In the situation below, R2's third out is a timing play. Even if the bases were loaded, the batter's fly out removed any force plays. Since R3 scored before R2 was declared out on appeal, R3's run counts.