As for being variable, probably not outside of the margin-of-error. If the ship was full of goods, then it would be heavier, and would sit lower in the water adding more resistance to forward motion. So the weight would have an effect, but whether it's noticeable outside the errors introduced when measuring, I don't know.

It's up to the Chief Engineer if he/she wants to use an Excel spreadsheet to track fuel consumption (and most probably do). As for tracking it based on weight, maybe some chiefs do that, but I imagine it's not worth it for most.

I wasn't a chief, so I don't know what their responsibilities are in regards to fuel consumption. But I imagine they would have to give estimates of future fuel needs to the company based on the ship's schedule.