If a weapon, e.g. the Shishkebab/ Gehenna, does fire damage, is the whole weapon damage increased by 50% or solely the fire damage part? So in case of the Gehenna 5 fire damage for 5 sec it would be +2.5 fire damage for 5 sec?

It increases anything that has initial fire damage and DoT, the only one that doesn't do this is Cleansing Flame, as it increases it's DoT through your Energy Weapons skill.

So basically, Gehenna goes from 42 to 63 just from pyromaniac. Hope it helps.