For Melee Override skills such as Maya's Scorn, do melee damage enhancements like badass ranks, melee damage skills, melee mods, melee relics and +melee damage weapons increase its damage?
Override skills such as Scorn are not affected by the following +Melee Damage bonuses:
(All tests done as a Level 50 Maya with Scorn; Scorn dealt 722 damage per beam on every test. Melee damage Badass Ranks and +% Melee Damage talents untested as I have no available Tokens and Maya has no talents with +% Melee Damage.)
TLDR: Yes, for Zero, Gaige and Salvador. No, for Maya and Krieg.
Scorn (Maya / Siren)
Although thrown with the melee button, Scorn is more appropriately considered a slag attack. It does not benefit from effects that directly improve melee or weapon damage, but does benefit from improved slag damage (i.e. Elemental relics). Strength relics will not improve its damage, but will improve cooldown if they possess a +Melee Override Cooldown Rate ability.
Execute (Zer0 / Assassin)
As it counts as a Melee Override ability, even if a bayonet / bladed weapon is equipped, performing an Execute will display a sword slash animation (slightly different from the normal melee sword slash). The melee damage bonus from the bladed weapon still applies.
With Claws (Gaige / Mechromancer)
This skill activates when bladed weapons are used but only overrides the animation. The melee attack still receives the bonus melee damage from blades, roid shields, etc.
With this skill, each stack of Anarchy adds 0.6% to Gaige's melee damage, up to a maximum of +90%, or +240% with five ranks of Preshrunk Cyberpunk. This damage can be considered doubled, since each melee attack used with Anarchy hits twice in rapid succession. With the Slayer of Terramorphous Class Mod, this goes even further, up to a maximum of +360% Melee Damage.
Fistful of Hurt (Salvador / Gunzerker)
Fistful of Hurt quadruples the damage of a melee blow. This multiplication (×4, or +300%) takes place after every bonus from relic, skill, or wielded weapon is applied.
Hellfire Halitosis (Krieg / Psycho)
Like Maya's Scorn, Hellfire Halitosis is not affected by bonuses / effects that directly improve melee or weapon damage, but does benefit from effects that improve fire damage. Strength relics will not improve its damage, but will improve cooldown if they possess a +Melee Override Cooldown Rate ability.
Those damage boosts do effect melee override skills; you simply tested with a poor skill choice. That overrides the melee attack with a NON-melee attack.
I am certain the Mechromancer claws are effected by weapon blades; in my times I have used only melee and hide of terra against enemies. Roid rage damage certainly effects it too. I have seen the same results with the Gunzerker uppercut, effected by all the melee boosts that effect normal melee attacks.
Otherwise, the melee override for Zer0 would basically be the worst-spent skill point ever, right? It would negate any melee bonuses he had as long as he was attacking from deception.