- Yes, but it has nothing to do with flyby attack
- That seems reasonable.
When you have the grappled condition, you can't use your normal movement:
Grappled creatures cannot move
AFAICT the only exception is to move as the result of a successful grapple check. In that case the creature can make the check, and move 20 feet, and it can do this whether it has flyby attack or not.
Further, it doesn't actually use its move action to do this; it's part of the grapple action. You're moving instead of dealing damage, pinning, or any of the other possible grapple actions.
There are some corner cases where you could use grapple in conjunction with fly-by attack, but you'd have to move, grapple, and then drop the grapple as a free action to continue moving.
A creature with grab could use it to avoid the grappled condition, and thus be able to move freely. But I don't think they could take the creature with them, so you'd have to stay within reach.
You're in luck, 'cause there are actually rules in Pathfinder for cave-ins and collapses:
Characters in the bury zone of a cave-in take 8d6 points of damage, or
half that amount if they make a DC 15 Reflex save. They are
subsequently buried. Characters in the slide zone take 3d6 points of
damage, or no damage at all if they make a DC 15 Reflex save.
Characters in the slide zone who fail their saves are buried.
Characters take 1d6 points of nonlethal damage per minute while
buried. If such a character falls unconscious, he must make a DC 15
Constitution check each minute. If it fails, he takes 1d6 points of
lethal damage each minute until freed or dead.
The given CR for this sort of environmental hazard is 8, but I'm sure you'll be able to scale it up or down, depending on the party level and your intended deadliness of the encounter :-)
Edit:
Going deeper, as required by Eric B :)
On escaping the collapse, let me keep quoting the paragraph I linked to:
Characters who aren't buried can dig out their friends. In 1 minute,
using only her hands, a character can clear rocks and debris equal to
five times her heavy load limit. The amount of loose stone that fills
a 5-foot-by-5-foot area weighs 1 ton (2,000 pounds). Armed with an
appropriate tool, such as a pick, crowbar, or shovel, a digger can
clear loose stone twice as quickly as by hand. A buried character can
attempt to free himself with a DC 25 Strength check.
But how does the underwater situation come in to play? Apart from drowning, we have:
Digging people out: Because water is denser than air, objects submerged in water will have higher buoyancy than when in air. Thus moving rubble and rocks should be easier under water. However, unless our heroes are affected by freedom of movement or similar enhancements, movement is down to one quarter (assuming successful swim checks). Right off the bat, I'd say moving loose rocks underwater can be done at roughly the same speed as above water. Few large rocks would decrease the time, and a huge pile of rubble would probably take quite a bit longer to move under water than above.
Freeing yourself: Because stuff is easier to mover under water, I'd give the submerged character a +5 bonus on the strength check to free himself.
Damage from the collapse: Since damage dealt underwater is halved, I'd apply the same 50% discount to damage from falling rocks.
Best Answer
From the wording of the description it seems clear to me that you can burst the rope while tied up.
The important distinction to make is between escaping the bonds and bursting them. You escape the bonds by passing the DC 20 + CMB of tying creature check. You burst the rope (and thereby are untied) by passing the DC 23 Strength Check.