Masamune gets it! The answer is, more precisely, the melody (or motif?) from the main theme (also used in Highwind). Here is the line-by-line:
Find me by land, or much later by sky
Of course it plays when you’re walking around (“FFVII Main Theme”), or flying (“Highwind Takes to the Skies”). However, it doesn’t play while flying around until you get Cloud back, hence the “much later by sky.” (You can hear it inside the airship, I know.)
When doom is at hand, the first option will die.
Once Meteor has been summoned, the main theme doesn’t play “by land.”
Twice I'm there when you've escaped via rope;
The first time is after the Sector 7 plate crash. It plays “On That Day, 5 Years Ago” in the scene immediately following--where Barret’s losing it and shooting at random. The second time is after the motorbike chase, when you use a rope to descend from the highway (“Holding my Thoughts in my Heart”).
Once while unconscious and regaining hope.
“Who Am I," the final variation on the motif, plays when Tifa and Cloud are in Cloud's "dream world".
It should be noted, I've more than one form--
Five, in fact, as you can see above. Also, "noted" is a pun about the nature of the answer.
Sometimes I'm chilling, but usually I'm warm.
“Who am I?” can easily be considered chilling. Warm refers to the normal melody in major, which is what you hear most of the time.
If you are dextrous, then carry me well
I’m sure you’ve all heard the phrase “to carry a tune”? Yeah..not an object. In order to carry a tune well on piano, you have to be dextrous (good with your fingers).
To coax divine pow'r from an old corpse's shell
…To get Tifa’s ultimate limit break, Final Heaven, from the piano in her house. The “shell” part of a piano is traditionally made of wood (much like a witch ), so of course the “old corpse” is a dead tree.
I know that some of the riddle was misleading, but just imagine if I'd replaced the word "find" (in the first line) with the word "hear." You'd get it in a snap! Of course, misleading you with the word "carry" was intentional.
Masamune's got the next one!