If another effect modifies how much damage a source would deal to an opponent or a permanent they control, including preventing some of it, the player being dealt damage or the controller of the permanent being dealt damage chooses an order in which to apply those effects. If all of the damage is prevented, Mechanized Warfare's effect no longer applies.

The additional 1 damage is dealt by the original source of damage. The additional damage isn't dealt by Mechanized Warfare.

If damage dealt by a red or artifact source you control is being divided or assigned among multiple opponents and/or permanents opponents' control, divide the original amount before adding 1. For example, if you cast a red spell that deals 4 damage divided among any number of targets and you choose to have it deal 2 damage to a creature and 2 damage to that creature's controller, that spell will deal 3 damage to each of them instead.