If hypothetically speaking, Sega had been able to stay in the console race, what do you think their 7th gen followup to the Dreamcast would have been like, and why?

It would have been a variant of this, the SEGA Lindbergh. Virtua Fighter 5. Afterburner Climax. Many more 360 era ports were from this arcade machines. Much like how Dreamcast was a cutdown NAOMI. 7th gen SEGA console be a version of this arcade machine.  Personall I always saw 360 as the true Dreamcast 2. First Xbox as DC 1.5. Controller to online. Looks. All stem from DC. 

https://segaretro.org/Sega_Lindbergh

I am a huge Genesis/Dreamcast fan but... sad to say, I think it might have been something a little more digital than we would have cared for. I mean, this is the company that tried to do Game Pass via cable TV in the mid 90's, for crying out loud. They pushed forward, sometimes too far forward. So my guess is we might not have been too keen on whatever it was.

Or maybe we would have.

It's fun to think about.

Wow. I've never thought about this.

How would it be? Magical :D

It's hard to imagine but I the more I think about it the more I imagine a lot of niche Japanese games that would have been able to exist and find a Platform that wouldn't have been available otherwise. I wonder if they would have went the power route of the PS3/360 or a innovative yet cost effective route like the Wii (or maybe a very interesting middle ground). Possibly a Wii U style of device as that would have been a follow-up from the Dreamcast memory-screened controller.

and Sonic Adventure 3 (instead of Sonic 06 lol).

Kinda doubt they expand on VMU past DC. VMU was thought up in the middle of the Tomagatchi craze. When DC released in japan in 1998 that was still hot. 1999 release in the US. It had cooled down I believe. However DC did link with Neo Geo Pocket similar to what Gamecube did with GBA. So if they released a handheld in 2003/2004 might have expanded link abilities. A controller with a screen in 2005? Don't think that was a viable cost effective option. Wasn't really even in 2012 with Wii U. 

Sega does what Nintendon’t…

BLAST PROCESSING 2.0!

In this hypothetical scenario, how well did the Dreamcast do? I think a lot depends on that

For the sake of the hypothetical, let's say it stuck around but still kinda got steamrolled, kinda like Gamecube.

I think such a scenario would have required that Sega didn't botch the launch of the Saturn. It releases on the originally-planned "Saturnday" release of Sept. 2 and only costs $300, same as the PlayStation. Hard to say how well it does in such a scenario seeing as we can't rewind time or go to an alternate universe where that was the sole divergence point in history, but let's assume it does a lot better and doesn't bomb out of the gate, giving Sega enough of a user base to keep making consoles after the Dreamcast, which sells at least on par with the GC & Xbox.

It is interesting to think about how things would pan out if certain decisions were made differently. Sega avoids their mid 90s screw-ups. Nintendo stays partnered with Sony and releases a hybrid cartridge/CD system in Gen 5 (basically something like the original "Nintendo PlayStation" plan). Nintendo and Sony still split and the PS1 still releases, but Nintendo decides to make a disc-based or disc/cart hybrid N64. The PS3 ends up being a system that only cost $400 at launch. Microsoft doesn't screw up the lead-up to the Xbox One. Instead, we got the results we got, and those were some of the biggest industry-shaking events ever. The only thing more industry-changing was the Crash of '83.