Your DM is wrong in your specific case. I'm not sure whether you're saying delaying doesn't end a sustain -- it does -- so I'll cover readying first, then sustaining.
Readying is covered on page 247 of the Rules Compendium and page 291 of the Player's Handbook. Your reading is correct. You would have to sustain before you spent your standard action to ready, but on the next round your initiative would move as per your understanding, and you'd sustain at the normal time. This is in fact "free" sustaining; however, it's enough of a fringe case so that it's never bugged me. On the other hand, I can see it as a house rule. As a GM, I'd just ask people not to be abusive if they started metagaming around it, though.
Delaying is covered on page 242 of the Rules Compendium and on page 288 of the Player's Handbook. When your initiative comes around, your start of turn effects happen no matter what, even if you delay. Your end of turn effects are split into two parts. Any beneficial effects end immediately when you delay, and anything you were sustaining ends immediately. So yeah, your turn is moved later in the round -- but you still lose your sustains, and you don't get that free sustain for a portion of the round.
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:
- Alice starts falling on Bob's turn immediately after the trigger that causes her to fall
- 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:
- Alice starts falling on Bob's turn immediately after the trigger that causes her to fall (same as in the RAW case).
- 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.
Best Answer
Your example is somewhat flawed. If Bob readies an action to attack Alice as soon as she's within reach, and then moves within reach, it's just a normal move/attack sequence. No Readying is necessary, in which case Alice resolves her readied attack as soon as Bob is within reach and then Bob acts.
Let's go with another example.
Alice has the best initiative. Alice readies an attack if a foe enters a certain area. Bob has the next best initiative. He readies a spell if a foe enters the same area as Alice is watching. Then a foe enters that area. Who goes first?
The description of the Ready action (p. 192, PHB) doesn't state this situation specifically but it does imply some things:
First, the Ready action is used to "get the jump" on an opponent that is acting after you in the initiative round.
When the trigger condition is met, you get to go before your opponent because you would have gotten to go before them in the initiative order anyway.
If we take that and expand it to others who might be reacting to the same trigger, then multiple readied actions should be resolved in the order in which their characters would have normally acted in the initiative round.
So, Alice would go first, then Bob, then the foe if it survived.
So, my interpretation is that multiple Readied actions are resolved in initiative order (whatever that would have been).