Xanather's Guide to Everything now clarifies this in two ways. Firstly, it provides an explicit explanation of the RAW and provides an optional rule for long falls.

RAW - Immediate fall & immediate damage

The rule for falling assumes that a creature immediately drops the entire distance when it falls. (XGtE)

So by standard rules the answer to your questions are:

1. Alice starts falling on Bob's turn immediately after the trigger that causes her to fall
2. She immediately falls the entire distance and takes the damage (barring some intervention that halts her fall).

As for your follow up questions:

• Is a reaction (e.g. casting Feather Fall) on Bob's turn the only hope?
  • That or a readied action are about the only ones I can think of right now
• Can Alice's ally Charlie swoop in on his subsequent turn within the same round to catch her?
  • No, she has already fallen
• Can Alice spend half her movement on her next turn to "stand from prone" and pull out of her involuntary dive?
  • No, she has already fallen
• If Alice had expected to be shot down, could she have spent an action to be Ready to "stand from prone" while moving through Bob's airspace?
  • Yes, you can prepare an action this way to use half your movement to stand up from prone.

Optional RAW - 500ft/turn & End of her next turn

However, XGtE also provides an optional rule for long falls that says:

When you fall from a great height, you instantly descend up to 500 feet. If you’re still falling on your next turn, you descend up to 500 feet at the end of that turn. This process continues until the fall ends, either because you hit the ground or the fall is otherwise halted. (XGtE)

In this case, the answers to your questions are:

1. Alice starts falling on Bob's turn immediately after the trigger that causes her to fall (same as in the RAW case).
2. She falls 500 feet instantly which still leaves her 105 feet in the air. On the end of her next turn she will fall the rest of the distance if she does nothing, but allows her to presumably use her turn to take preventative action if she so desired and was able.

As for your follow up questions:

• Is a reaction (e.g. casting Feather Fall) on Bob's turn the only hope?
  • No since she will take her next turn (as would anybody else who was there) before hitting the ground
• Can Alice's ally Charlie swoop in on his subsequent turn within the same round to catch her?
  • Yes! Though the mechanics of how this would actually work would be up to the DM as catching a falling person is going to not be an easy task!
• Can Alice spend half her movement on her next turn to "stand from prone" and pull out of her involuntary dive?
  • Yes! At the beginning of Alice's turn per:

    But if that creature starts any of its later turns still falling and is prone, it can halt the fall on its turn by spending half its flying speed to counter the prone condition (as if it were standing up in midair). (XGtE)

• If Alice had expected to be shot down, could she have spent an action to be Ready to "stand from prone" while moving through Bob's airspace?
  • Yes you can. Same as with the standard RAW.

Which method is best for your group will be up to your DM, but personally, I find the optional rule to be much more intuitive and reasonable an answer at my table.

No, a rolled death save will not spawn a zombie - Finger of Death can kill in 2 ways

Way 1: Massive Damage

Finger of Death deals damage greater than the targets Current HP+Maximum HP. The target dies instantly from massive damage and rises as a zombie

Way 2: 3rd Failed Death Saving Throw From an FoD hit

The target is unconscious* at 2 failed death saving throws. FoD hits causing an instant failed save (from damage while unconscious). The target dies and rises as a zombie.

Finger of Death is only the (mechanical) cause of death if you die while the action is happening and because of that exact action.

*One possible exception is on a Zealot Barbarian who delays the death due to raging but ends their rage having been hit by FoD on at 2 failed saves. The delay would occur but they would become a zombie as soon as they truly die (If they are still at 0 hp at the end of rage, and have not been healed since FoD "killed" them).