Let's say for a moment, that Sega in 2001 had massive cash reserves, and decided they would continue to bankroll the Dreamcast rather than abandoning their hardware business.

How do you think the rest of the 6th gen would play out in this scenario, and the years that followed? How would gaming history change?

Similar result, just different date. You need to change the fate of the SEGA Saturn to save the Dreamcast. To do that, you need to never do the 32X and reorganize SEGA's upper management so there isn't so much dysfunction between Japan and America. The only reason SEGA didn't die itself is that the former CEO, on his deathbed, donated his entire fortune to the company to prevent it from going out of business. SEGA was bankrupt. Dreamcast wasn't oh this failed so we quit. It was years of failure and losing money. Unless Dreamcast took off like the PS2 and set record numbers. SEGA and DC were never going to survive.

Might have moved the needle a little bit among GC, Xbox, and Dreamcast, but in no universe are they able to overcome the dominance of the PS2. If Microsoft and Nintendo, who both did have massive cash reserves and name recognition that far surpassed Sega, couldn't do it, Sega certainly wouldn't have.  The Dreamcast would also have started showing its age compared to its competitors.

The issue of overcoming the Saturn's disastrous performance would have sitll remained. The Xbox is basically what you described - not blindingly successful and a complete bust in Japan, but bankrolled by a megacorp that kept pouring money into it because they feared that the PS2 might hollow out the home PC business.

Sega's favorite pastime has always been to throw away money and opportunities with their bad decisions. If Sega had massive cash reserves they wouldn't have been Sega.

A reminder that back then consoles swiftly got price cuts and sold most of their units at lower price points later on. Sega had great console launches but always dropped the ball when it came to the time consoles really flew off the shelves.

Let's suppose however they continued, perhaps by striking a deal with Microsoft or something to make up for the loss of EA's support. There's a very large chance that the Dreamcast 2 would've resembled the Xbox 360 a lot in both features and time of release since the OG Xbox already picked up on a lot of what Sega was working on back then. It would have seen massive success and put them in their best position since the early 1990s.

No Xbox and yes Dreamcast would also have changed Nintendo and Sony's plans in unpredictable ways beyond the GameCube and the PlayStation 2. The next Sega console would probably have looked a lot different than the Xbox One since that was mainly designed by a lot of Microsoft folks who had nothing to do with the OG Xbox or the X360.

If Sega had large cash reserves and better negotiating tactics, they could've had a much better relationship with Microsoft and convinced them not to enter the race with the original Xbox.
Dreamcast survives until around 2004-2005 and we get a Dreamcast 2 or whatever it would be called that is similar to Xbox 360.
Some of Sega's software from our reality would still get made. I think Sonic Heroes and Shadow the Hedgehog might've still happened and they'd probably be Dreamcast games or at least Heroes would be. Shadow might be saved for Dreamcast 2.
They would have to have crazy cash reserves to get past the seventh generation because I don't even know how they would be able to be successful every generation. I really can't fathom how their eighth-generation console in this situation would be successful because they probably would run out of steam by then.

I wouldnt been able to play Sonic Adventure 2 Battle which is my favorite Sonic game if that happened.

I would imagine Sega still leaving the console business, just in 2004 or 2005. Which would still end up being a big deal.

• The Sonic fandom would likely be a very different one. No handheld 2D entries, we'd get Adventure 3 instead of Heroes, and retro compilations would likely be made for Dreamcast and PC instead of GameCube. Note that Sonic and Sega were the biggest third party hits on GameCube, so the platform would likely be badly hit.
• GTA3 MIGHT have still been released for the Dreamcast, as it was in development for that platform at a point before shifting to PS2. However, I don't think the series could have stayed on Dreamcast with the limitations of the GD-ROM.
• Sega with money would have kept a lot of the staff that left to form other studios. These studios included Artoon/Arzest, Q Entertainment, etc.
• The Xbox would be badly hit without games like Ninja Gaiden, Jet Set Radio Future, Panzer Dragoon Orta, Shenmue II, OutRun 2, etc.

Play SEGAGAGA to find out how it goes ;)

So many ifs'

If sega put some money forward this would not have been cancelled and could have boosted the system 

A few decades late...

But GTA3 is being ported to Dreamcast.