Some questions about "specific" Damage Conversion cases

Hi! its my first time here and i have some questions, thanks.

First of all, my english isnt good, sorry.

1-)
The "100% increased Melee Physical Damage against Ignited Enemies" from the "Razor of the Seventh Sun" is used if i convert my damage? If i have 100% physical do fire for example, my fire damage will be increased because of the "100% increased Melee Physical Damage against Ignited Enemies" mod? (when i hit with the weapons, of course)

2-)
If i have "hatred"/"herald of ice" active when i use a melee physical damage skill linked with "Cold to Fire Support", the cold damage part dealt by the skill will be 50% converted to fire? Or i need some global "Cold to Fire" to convert in this case?

3-)
"Added Fire Damage Support" works with skills that deals 100% converted physical to elemental?

Wiki says this: "Added Fire Damage is a support gem that adds fire damage to supported skills based on the skill's physical damage."

But also says this: "The amount of fire damage added is a percentage of the physical damage dealt by the skill and follows the rules for damage conversion."

As long as the damage was originally physical before being converted, all modifiers to physical damage work on it.

so yes, yes and yes.

Thanks!