Thornmail is also triggered by Infinite Duress, so it would appear that Thornmail applies to anything that would apply life steal.
Damage is reflected when hit by an attack, regardless of whether the attack deals damage. If the attacker is blinded, or the wearer is protected by a shield or invulnerability, damage will still be reflected.
Abilities that apply on-hit effects, such as Blade Waltz, will trigger Thormail.
If you have >233.3 effective armor, Thornmail will reflect more damage than you receive, assuming the attacker has 0 magic resist; if they have 30 magic resist, >333.3 armor is required.
Thornmail is a particularly effective defensive option when facing down opponents whose damage is dealt primarily through autoattacks, or against opponents that utilize on-hit healing to survive fights (such as life steal).
There is a common misconception that life steal and magic resistance "counter" Thornmail, but this is largely incorrect. A combination of magic resistance and life steal will mitigate the reflected damage, but only large quantities of either will see net health gains from life steal.