Tagging Up after Ball Thrown Out-of-Play

 
With one out and runners on third base (R3) and first base (R1), R1 is stealing on the pitch. The batter flies to right field, where the ball is caught for the second out. After the catch, the right fielder throws towards first base where he could easily have doubled up R1 (since R1 was stealing and was far away from first base). However, the right fielder air-mails his throw over the first baseman and out of play. R3 had properly tagged up, but had not yet crossed home plate when the ball went out of play.
 
The umpire calls "time' and properly awards R1 third base (two bases from the time of the right fielder's throw) and awards R3 home. When R1 arrives at third base, the defensive coach goes to the umpire and thinks R1 should have gone back to first base to tag up before he advanced to third base on the award (R1 did not return to first base to tag up).
 
A) Is the defensive coach correct that a runner must properly tag up even when the runner is awarded bases?
B) If R1 is called out for not tagging up, this would be the third out of the inning. Does R3's run count?
 
Call
Runners must run the bases properly/legally -- or be at risk to be called out upon proper appeal. This is true for runners advancing on batted balls as well as when bases are awarded.

In the situation below, R1 did not properly re-tag after the ball was caught by the right fielder. Even though the ball was thrown out of play (and R1 was awarded two bases), R1 must complete his base running responsibilities properly/legally. Since R1 did not and it was properly appealed, R1 is called out for the third out.

This is a timing play. R1 was called out well after R3 scored, so R3's run counts.