Which factors influence the gold gained from mercenary quests?
Solution 1:
Merc gold sure seems to be based on your Timelapse gold, which is based on how much gold you'd get in 8 hours of farming your HZE in a given Ascension. That takes into account your current DPS and what gold ancients you have. But it also isn't perfect, because it applies idle bonuses to that total even if you would need to be active to do any damage whatsoever. To the best of my knowledge, no one had figured out the exact equation. And that's not because it's a particularly difficult problem, it's because it really doesn't matter in efficient play.
The people who are both smart enough and dedicated to CH enough to figure these things out are also primarily concerned with efficiency, and introducing idle ancients into a full active playstyle to sporadically collect Merc Gold quests to push deeper into an run you almost certainly should have already Ascended out of isn't efficient. It also isn't wrong or bad, and anyone who tries to make you feel bad for playing any (solo) game the way you want can sod off. Disclaimers aside, most players only bother with Merc gold quests during their FANTs, to be used right before respecing all HS into Solomon at zone 104. As long as the gold gets you to Frostleaf 50+, you can get to 130, the last guaranteed primal boss and an ideal zone to ascend. And by the time you're regularly using TimeLapses, you're doing it to skip zones on your LABT, and the gold isn't really a concern. In either case, calculating the exact amount of gold you'd get is unnecessary.
You might consider visiting the Clicker Heroes Subreddit, specifically in the Monthly Help megathread. Someone might be interested enough to figure it out, or know someone who calculated it for their own satisfaction without ever bothering to post it.
You also might consider switching your calculator to Hybrid, and using the Nogfish trick, which is taking all your clickers off everything, waiting for your idle bonus to kick in, then clicking the orange fish, netting a huge gold spike.
Sorry this isn't quite the answer you were looking for.
TL;DR: No one knows.
Glossary of terms
HZE: Highest Zone Ever
FANT: First Ascension of a New Transcension
LABT: Last Ascension Before Transcension