Mail is basically mostly vulnerable to two types of attacks - pinpoint (piercing) and blunt. But It's not really possible to decapitate anyone with a thrust or blunt trauma. The video, with Peter Woodward, made horrible tests against the mail. The dummy was fixated with a couple of twigs resembling body parts (even a child can break twigs with ease), and furthermore it wasn't properly wielded nor did it have a gambeson underneath.
The axe would most likely not harm the mail itself in a slashing attack, but possibly deal blunt damage to where it hit, of course it depends on the momentum of the attack but the mail is likely to hold. The mace attack varies much on how hard it hits, where it hits, and how the opponents body move with the weapon. If lucky the knight may get away with a bruise. As for the sword it pierces the armor, but take a look into the link Night Ninja posted and you'll see the real effect of those piercing attacks, it's quite different from a hanging piece of metal.