Seeing as it's gonna rain for the next several days my mind has wandered off into the minutiae.

Does the fuel source affect the metal heated in a forge?

Byproducts of coke versus heating oil vs propane?

Do either affect the metal positively or negatively depending on what the metal will be used for?

Only negatively if you over-heat the steel. Regardless of the fuel type.
There is no magical combining of the heat source fuel to the steel.
The only thing the forge is used for is as a heat source.
The most important aspect is the ability to create and control a fuel rich environment (reducing atmosphere) as opposed to an oxygen rich environment (oxidizing atmosphere.)
If the super heated oxygen consumes all of its fuel while still in the forge environment, and there is still un-burned oxygen left over, it will begin looking for the next available fuel source - and that's your steel.
You want all of the oxygen consumed in the combustion process, leaving un-burned fuel. That way your steel is not part of the combustion process.
When you see a propane forge running and there is a "dragon's breath" exiting the forge opening about 6 inches, that is un-burned fuel that did not have enough oxygen INSIDE the forge to burn, so just as it exits the forge body it uses the oxygen right outside the forge to ignite.
That's a good thing.

This is a picture of my burner assembly which runs my vertical forge.
The gate-valve regulates the amount of oxygen and the needle valve regulates the amount of fuel.
Depending on how hot I want my forge, which I will control by line pressure from a regulator, I can independently adjust the air/fuel ratio to maintain my "dragon's breath".

At the 9:00 mark you can see the Dragon's Breath to which I refer:

Damascus forging