So... Did Jen commit suicide or not? I thought that considering what we've seen her do before (flying, etc.), she probably didn't actually die. Lo's smile after she jumps should also be noted. I've seen multiple interpretations of this scene, but I prefer to think that she lived because it would otherwise be the biggest Downer Ending ever.
This seems to be a point that Ang Lee has left deliberately vague in the film, and has spawned multiple interpretations. The most convincing explanation this troper has come across (see http://www.brightlightsfilm.com/33/crouchingtiger.php ) is that Jen, in her quest to be a free warrior, has seen that this path has not brought happiness to the other women fighters — neither to unscrupulous Jade Fox, nor the honourable Shu Lien. She realises that no matter how much Lo loves her, and even if he were to marry her and cherish her, true and honourable freedom cannot exist for her as long as she is a woman in that world. The secret wish she makes before she jumps off the mountain is to be reborn as a man — or perhaps as a woman who can truly be free.
The finest explanation I've ever heard still comes from my Dad: It's that Jen has already fallen for Li Mu Bai at that point and her jump accompanies a wish that he would come back to life for her again. For hints of this, watch how Jen's eyes light up when she gets Shu Lien to talk about Li Mu Bai early on in the film. She's obsessed and swooning over the man and his reputation. Over the film though the fact that Li does not return her affections, and there's this crucial exchange at the waterfall fight, when he said: "I took her out here so I could see your true self." To which she replied: "What do you KNOW about a true heart?" as a reference to not just how he has suppressed his desires all these years, but that he isn't getting the affection she has for her. This is why Jade Fox sets up Li and Jen in the trap at the end when what she has drugged Jen is with is an aphrodisiac ("Is it me or the sword you want?") and then lured Shu Lien there, hoping to destroy their relationship forever by luring Li into a trap. Li finds it in himself to resist and behave heroically and this finally breaks Jade Fox and results in his death. Shu Lien's advice to Jen to be "true to herself" prompts her to respond to it by jumping off for the one she loves... Which is no longer Lo. It is this moment that Jen learns the true meaning of heroism: It's not being able to triumph over others physically, as she has thought throughout the film, but it is giving your all for the one you love.
Or Jen was so overly impassioned and realized she could never be happy with all the options this life had to offer her; since the security of her life in an arranged marriage would leave her starved of adventure, but all her attempts at adventuring turned sour through deceit and desperation. I believe she attempted to commit suicide and/or flee by jumping off the waterfall bridge because she realized she didn't have the more wherewithal to be a truly honorable and respected warrior.
Considering how wobbly it is, would the Green Destiny even be remotely effective in a real fight?
Yes. This isn't the right forum for full details, but Tai Chi, at least, accommodates a flexible blade handily. Presumably other styles and methods do as well.
Yes... And no. The Chinese Jian sword (of which the Green Destiny appears to be) has a flexible blade; however, the stuff you usually see on demonstrations uses a thinner blade so it "wobbles" more than a "military grade" jian. It's still going to kill you dead; it's just not what a soldier would generally use. The forms and technique are the same, though.
Clearly the Green Destiny is thin enough that it wobbles just from being waved around. A good sword shouldn't flex until you actually press it against something hard. From what I've read, the reason the blades of those swords used in Wushu demonstrations are so thin and weigh very little is so that the performer can wave it around more easily and do fancy tricks that a Wujian wouldn't be light enough for. It also makes a cool swishing noise and the vibration of the blade looks flashy. However, they're almost more like sword shaped props than actual swords. Such an underweight blade would have difficulty cutting through sinew and bone as opposed to just slashing the surface; while sharpness is important, it is just as important for the blade to have some actual mass and stiffness without being too heavy either. An excessively flexible sword might bend like a spring instead of penetrating if you thrust an opponent wearing thick garments, and if there is less material in the blade it will be more easily damaged by weapon-on-weapon parrying, especially if your opponent is using something heavier like a wudao (saber) or a glaive. If it were a realistic movie the blade of Green Destiny would have been wrecked by defending against the arsenal of weapons Shu Lien used against Jen, but in-universe it's a magical sword so the edge isn't even nicked while all the regular weapons are destroyed.
This is more regarding the second movie than the first, but given that the heroes don't want to use the Green Destiny and just want to keep the villains who'd use it for evil from getting it, why don't they just drop it into a deep lake or melt it down in a forge?
It belonged to Li Mu Bai's master: a hero's hero as it were. Destroying it would be a pretty disrespectful act.