The Warring Triad's last ability isn't a mana ability. It uses the stack and can be responded to.

As The Warring Triad enters the battlefield, the number of cards in your graveyard will determine whether any replacement effects that affect creatures entering the battlefield apply to The Warring Triad.

The type-changing ability that can make The Warring Triad not be a creature functions only on the battlefield. It's always a creature card in other zones, regardless of how many cards are in your graveyard. It's always a creature spell while it's on the stack.

When The Warring Triad enters the battlefield, the number of cards in your graveyard will determine if a creature entered the battlefield or not for abilities that trigger whenever a creature enters the battlefield.

The Warring Triad's abilities function while it's on the battlefield, regardless of whether it's a creature.

If The Warring Triad stops being a creature, it loses the type creature and the creature type God. It continues to be a legendary artifact.

If The Warring Triad is attacking or blocking and it stops being a creature, it will be removed from combat. It won't rejoin combat if it resumes being a creature later during that combat.

Counters put on The Warring Triad remain on it while it's not a creature, even if they have no effect.

If an effect causes The Warring Triad to lose all abilities, its ability that causes it to stop being a creature still applies if appropriate.