A simple "non-spoiler" list of what triggers what changes in the game and environment. Quest names will be spoiled but not the context or requirements to complete said quests.

Completing "The Call of Cash"
Completing "Shop Till Drops"
Completing "Shop Till Drops"
Opening ANY backpack after starting "Hot Dry"
OR
Completing "Hot Dry"
Completing "Cleansing Fire"
Completing "Light It Up"
Watching cutscene after completing "Light it Up"
Watching cutscene after completing "Light it Up"
Time passing in-game after completing "Golden Cage Breakout"
Pay back your debt and complete either "The Vault of Truth" and/or "Operation Black File"

Yellow quests follow their own rules and are curated, so the contents of the cash are too!
There is a little bit of variation in the amounts for these, but overall it's going to be roughly the same.

All 20's and 50's are marked. Roughly half of the 100's are marked. There is enough non-marked 100's to complete the quest. All marks are Gang-marked bills, but each denomination belongs to a different gang. Neat!
Only the bag with the necklace has marked bills. And they are all Gang-marked bills
All boxes are mixed with Police-marked bills
Entire order is riddled with over 500 FDI-marked bills and some Gang-marked bills
Half of the order is Gang-marked bills, specifically from the Cartel
This one is interesting! So if you bother to separate the loose bills from the packs in this, there are a few FDI-marked bills among the loose cash. Otherwise the marks here are over 1000 Police-marked bills
The utter boat-load of marks! This Quest has it all! over $70k in Gang-marked bills, over 3000 Police-marked bills, AND over 3000 FDI-marked bills among this large quest!
These two aren't exactly "Yellow Quests" but I'm including them
The Vault is a little below the magnitude of Combo Heist. It has a couple Gang-marked bills, over 700 Police-marked bills, and an astounding over 3500 FDI-marked bills
This little nugget is available from a fresh game creation. It has, reliably: $33,500 in Gang-marked bills and 365 FDI-marked bills

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All 20's and 10's are fake. All 50's are Real. As of Latest Patch, the sports bag of 100's is all legit, the box is mixed fake and legit.
The box and 100's on the floor are all mixed Legit and Fake
Half the order is Fake
Half the order is Fake
There's like a couple fake bills interspersed in the big sea of legit and marked bills. Only in the XXL box and the sportsbags though. The dirty bank bags have no fake bills in them

Momma Jane has all the details. Spoilers ahead! Answering the question of what quests lead to what other quests!

There are 4 yellow quest lines. 6 if you count the two endings but I don't.
I am defining a "quest line" as a series of quests that directly correlate to another appearing.
Mandatory is defined as necessary to achieve an ending.

All of these quests need to be finished in order (*mostly) before you can unlock the 2nd room expansion into the main circular room.

• Fresh Steps for Mr. Fresh
  
• Matchmaker in Crime
  
• The First Delivery
  
• Tracking Quests
  
• The Call of Cash
  
• Finish The Real Order
  
• Shop Till Drops
  
• Clean it or skip it ** (Can be discarded)
  
• The Wet Case
  
• Fifty Shades of Cash
  
• Hot Dry ** (technically only have to start this, not mandatory to finish, but cannot cancel)
  
• Cleansing Fire

After this, the quest line splits in two ways, a continuation of quests that start becoming optional, and the other leading towards the story's 2 endings.

Most of these quests have increasing Rank requirements or odd requests so you will likely intersperse these with other quests.
Prerequisite: Finish Call with Betty after the fire

• The Light Test
  
• Light It Up
  
• Feeding The Pig
  
• [Rank 8] Europhoria
  
• [Rank 9] Blacklight Evidence
  
• [Rank 10] Saving Piglet
  
• [Rank 11] Golden Cage Breakout ** (Can be discarded)

After completing this list and passing time in-game, The Vault should drop in! Now pick your ending!

Prerequisite: Spawn the Vault.

• [Rank 10] Operation Black File
  
• Priming the Shot
  
• Loaded but Undecided
  
• Fill the Piggy **can be done at any time
  
• Point of No Return

Prerequisite: Spawn the Vault.

• [Rank 11] The Vault of Truth
  
• Launch Code: Oink
  
• Pre-Launch Protocol **Can be done at any time by finding the monitor
  
• Inflation Sequence
  
• Fill the Piggy **can be done at any time
  
• Final Ascent

None of these are required to get an ending but it is advised to do Clean Cut at least for the free washer. These all have low rank requirements and can all be done before starting the Room 2 Mandatory Quests. Additionally all but the first two can be discarded and all money kept.

• Clean Cut
  
• Two Cases
  
• [Rank 3] From Chaos To Cash ** (Can be discarded)
  
• [Rank 2] Silent Deal ** (Can be discarded)
  
• [Rank 4] Fake Fluff ** (Can be discarded)
  
• [Rank 2] No Stains ** (Can be discarded)

At this point the quest line will pause unless you have already unlocked the main circular room.

None of these are required to get an ending.
Prerequisite: Watch Cutscene to open Main Circular Room AND complete Optional Quest Chain 1.

• [Rank 10] Different Values ** (Can be discarded)
  
• [Rank 9] Luxury Wrapping ** (Can be discarded)
  
• [Rank 8] Fragile Treasure ** (Can be discarded)
  
• [Rank 9] Financial Exorcism ** (Can be discarded)
  
• [Rank 8] What they left behind ** (Can be discarded)
  
• [Rank 11] Financial Zoo ** (Can be discarded)
  
• [Rank 8] Tracing the Bullet ** (Can be discarded)
  
• [Rank 10] Collector's edition ** (Can be discarded)

New Spawn Logic As Of Latest Patch!
You've opened the vault, and there's a bunch of money and more Quests!! Combo heist might mix into the Optional Quest Chain pool

• [Rank 8] Combo Heist ** (Can be discarded)