Puncture

DoT cannot ever apply leech.
The additional damage while moving is additional damage, and is described as such on the gem and skill (unless something's been broken, but the gem linked earlier in the thread still has that for me). You're always taking the base 10% DoT, and while moving you're taking the additional 50%, for a total of 60%.

1) yes
2) no, culling strike applies when you hit the enemy, not at any other time
3) yes

Things that will directly affect bleed damage from Puncture (off the top of my head):

• Damage modifiers
• Damage over time modifiers
• Physical damage modifiers
• Projectile damage modifiers (if applied with a projectile)
• Area damage modifiers (if applied with an AoE)
• Totem damage modifiers (if applied with a totem)
• Trap damage modifiers (if applied with a trap)
• Mine damage modifiers (if applied with a mine)

There are projectile attacks and there are projectile spells. "Projectile" is not limited to certain kinds of skills, it's simply a method of causing damage that can be used by skills of any type.

Melee, however, is fundamentally a subset of attack. Melee is only defined in terms of attacking with melee weapons. There isn't any concept of melee outside of attacks in the game. Since DoT fundamentally cannot be attack damage, it also cannot be melee damage.