An ability that triggers when a creature becomes monstrous won't trigger if that creature isn't on the battlefield when its monstrosity ability resolves.

Monstrous isn't an ability that a creature has. It's just something true about that creature. If the creature stops being a creature or loses its abilities, it will continue to be monstrous.

Once you declare you're casting Nemesis of Mortals or activating its ability, it's too late for players to try to change the number of creature cards in your graveyard to affect the cost of doing so.

Neither the cost-reducing ability of Nemesis of Mortals nor the condition of its monstrosity ability can reduce the colored mana requirement of those costs.

Once a creature becomes monstrous, it can't become monstrous again. If the creature is already monstrous when the monstrosity ability resolves, nothing happens.