For Path of Mettle's last ability to trigger, any two attacking creatures you control need to each have any one of the four listed abilities. They don't need to share one of those abilities. For example, attacking with Sun Sentinel (a creature with vigilance) and Fanatical Firebrand (a creature with haste) will cause Path of Mettle's ability to trigger, as will attacking with two Sun Sentinels.

Path of Mettle's first ability deals 1 damage to each creature that doesn't have any of the four listed abilities. It doesn't deal 1 damage to each creature for each ability that creature doesn't have.

A creature with indestructible can be chosen at random. It won't be destroyed.

A creature with hexproof can be chosen at random. It will be destroyed.

A creature that attacked and was removed from combat (such as by Spires of Orazca) still attacked, so it can be chosen at random.

Players can't take actions between choosing the creature at random and destroying it.

In a Two-Headed Giant game, Metzali's second ability causes the opposing team to lose 4 life.

Metzali's last ability can be activated before blockers are chosen.

Metzali's last ability can be activated after combat damage has been dealt. You'll choose a creature at random that attacked this turn and has survived combat damage.

A creature that was put onto the battlefield attacking didn't attack, so it can't be chosen at random.