Game Situation: Many Runner Appeals

 
No out and no one on base. The batter (B1) grounds to the shortstop, who misplays the
ball but recovers and throws to first base. The speedy B1 beats the throw but clearly steps over first base without touching it. The base umpire signals "safe". B1 overruns first base but then returns directly to first base with both feet in fair territory. The defensive coach yells "tag him, he is in fair territory!", so the first baseman tags B1 before he returns to first base. The base umpire signals "safe" and B1 returns to the first base bag. The defensive coach then yells "He missed first base and was tagged before he returned -- he must be out now!"
 
1. Should the base umpire have signaled "safe" when the batter-runner clearly missed touching first base?
2. Is a batter-runner protected when he returns to first base completely in fair territory?
3. Should B1 be called out as a result of the defensive appeal?
 
Call
We have 3 Safes and 0 Outs on this play!
 
1. The umpire mechanic when the batter-runner beats the ball to first base (but the batter-runner misses the base) is to signal and verbalize "safe".
 
2, If the batter-runner overruns or over slides first base, he is protected from being called out if he immediately returns to first base (as long as he does not attempt, or fake, to advance to second base) NF 8-2-7. It does not matter if the batter-runner returns immediately to first in fair or foul territory (another urban legend debunked).
 
3. When the defensive coach appealed that the batter-runner had missed first base, the batter-runner had already returned to the base -- so he is safe.