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Q1: Yes, they will stack
Q2: No, because its a Heavy Weapon according to the game for perks, not an Energy Weapon
Q3: Yes, Heavy Gunner will add the damage for Heavy Weapons, including those that use Energy
Q4: Not quite sure of this one, but I believe no for the same as Q2, the game considers any and all grenades as explosive, regardless of damage type, so no
1. Yes, they stack, because dealing energy damage qualifies it for tesla, and being automatic qualifies it for Commando.
2. Yes for the gatling laser, since it deals energy damage. No for the gauss rifle or the railway rifle, since both of those deal ballistic damage (both are also rifle weapons, not heavy weapons).
3. Yes, since the perk only looks at whether the weapon is a heavy weapon.
4. No, grenades aren't affected, since they count as dealing explosive damage.
There's a bit more to how it all works behind the scenes of course.
The tesla armour property only cares if the weapon is dealing energy damage. The weapon can be an automatic, heavy, pistol, rifle, or even melee weapon. If it deals energy damage, it gets the tesla boost.
EDIT: And you can easily check each case by getting in and out of the Tesla armour and checking the damage numbers in the Pipboy.
Each piece of tesla armor adds 5% to the armor's energy damage multiplier.
(so all 3 = x1.15 energy damage)
Most multipliers multiply each other. (aside from ones that add to an actor value, in which case all the bonuses that add to that value are added up first, then the total is used as a multiplier)
So if you have Bloody Mess + Lone Wanderer (no companion) + whatever weapon type damage perk (Heavy Gunner, Commando, etc.) + 3xTesla Armor (using an energy weapon) you do: WeaponDamage x 1.15 x 1.25 x 2 x 1.15
Also, side note: there is a hidden explosive damage multiplier in Heavy Gunner (up to extra x2 explosive damage done by heavy guns) and Lone Wanderer (extra x1.25 explosive damage) in addition to their normal damage bonus.
Okay, now I feel dumb. This covers all my questions, thanks.