Basically my question is “what scales them?”
Forged weapons scale off all stats given by your weapon + any minion stats on gear (like minion damage or minion health). Most unique modifiers weapons won’t work unless they explicitly say minion. Best rule of thumb is if you can find the affix in the weapon section of the materials tab your weapon can use it otherwise you have to test if it works.
I would highly advise having % attack speed and % increased damage of whatever your building for your first 2 weapon affixes. Forged weapons have a 20% more attack speed bonus built in and get another 20% from the forge strike tree making them scale very well with attack speed. They also get a crap ton of base damage ( well over 400 if you weave in void cleave buffs), so you want increased instead of flat.
The smite damage buffs are self only, but the health and health regen are not.
If you put points into burning armor you would want to balance you gear and skill /passive points in a way that makes sure you have more flat fire damage than physical. That way your manifest armor will always to fire damage and all melee damage it gets from your weapon will convert to fire. % fire will then scale all of its damage rather than just the flat you have in the skill tree.