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.

Short Answer

As long as the item was gained within the rules of AL, there is no rule that supports a DM disallowing use of resources gained between sessions.

Long Answer

Suspension of disbelief is vitally important to organized play; it is almost a core precept. Characters are highly likely to run across other characters who have played the same content, but in a different order and with different players. Each player must disregard these discontinuities, because they are inevitable unless you never roam tables¹.

The DDAL FAQ (in the DDAL Player's Pack) provides explicit guidance under Leveling Between Sessions. The short version is: That's when you're supposed to do it - between sessions. In fact, this where the "suspension of disbelief" is explicitly mentioned. It doesn't just call out the players needing to do it, it calls out the DM need to do it, too. It also talks about new equipment, new magic items, and new class features.

When you're continuing a multi-session adventure, it instructs you to keep track of expended hit points, spells slots, and other consumables that refresh on a rest. If you're down 50 hit points and have used three first level spell slots left when you leave the multi-session adventure, you're down the same 50 HP and the same spent spell slots when you come back to it. You may have a higher HP total and additional spell slots because you leveled up, of course.

Specific Examples

#1

While there are story ties between the three modules in each trilogy, they are different modules as far as DDAL is concerned. They're one-and-done. They may make more sense if they're played sequentially and continuously, but there is no expectation that they will be. They should be treated like any other single modules.

#2

By the same token, each session stands alone. DDAL expects that characters will be used between sessions of the same hardcover. See "However..." below; the GM may be making an unreasonable request. In any case, he does not have the authority to deny you the use of properly earned and properly logged XP, GP, DT, Renown, or magic items. If you're following the DDAL rules, he should be, too. Those rules don't allow him to deny it, unless he's accusing you of cheating.

#3

Only example situation #3 is a "multi-session" adventure, and the passages referenced above cover that. Note that each "part" of the introductory adventures should be logged as an entirely separate adventure - because they are. They follow the same principles as I mentioned in #1 above.

However...

A DM could ask you not to play the character between sessions², but it's not anything the DM can demand under DDAL rules. It's just a request between two (presumably) friends. Whether or not this is a reasonable request varies by the situation - how long it's going to be between sessions, etc. What is acceptable in that group is a social contract question far beyond the scope of the original question here.

The DM always has the right to tell any player to leave the table. If the aforementioned request was polite and reasonable, and the player disregarded the request anyway, that makes the player the jerk. The DM is totally within his rights as a DDAL DM and a DM in general to give the disrespectful player the boot.

¹_{If you're never going to play the character at another table, not only is the original question moot, but why bother playing under DDAL rules anyway?}

²_{I always ask this of my weekly Hardcover players. Only one has ever disregarded the request, but he was at a convention and didn't have another T2 character to play in the content and schedule that was available to him, so I let it slide.}