Keeping in mind that a missed bag is assumed touched until the appeal. How does the live ball appeal work for missing the bag at 1B? My guess is this: B/R beats throw, but runs past 1B and misses orange bag. Throw arrives to F3, who is touching 1B. Is that the appeal--F3 catches the throw to 1B before B/R gets back to touch the missed bag? Or does F3 need to communicate something to umpire about an appeal while touching 1B while in possession of the ball?

JMHO, but in this case where runner beat throw, I would need some indication that
F3 is appealing that she [runner] missed the base. Appears that F3 has caught
a late throw, so now needs to indicate to me [umpire] that she is appealing the play.

It's gonna be a tough call for F3 as the throw is coming in and therefore cannot be looking at 1B. F3 would be getting a lot of help from other team members hollering TAG TAG! Makes it easier to call the appeal.

Not to hijack but to extend a bit. If lots of people(coach, other players) are yelling about the missed 1st base but F3 seems oblivious would you still grant the live ball appeal on the missed 1st base. This scenario happened to me the other night with F3 finally saying/looking to me just before runner returned. I was wondering what I'd call if there was no recognition from F3.

I would not. Unlike a dead ball appeal, a live ball appeal must be made with the ball by a player touching the base or player, and telling you s/he is making an appeal on something specific. To me that excludes the people without the ball and not touching the base or player, and excludes the player with the ball touching the base or player, but not making the obvious cognitive action.

What Steve said. Live ball, the only person who can appeal is the one with the ball and must still complete a tag or touch the base prior to the runner returning or the ball being killed.