I had a flat on the front right tire, and I've replaced that tire with a new advan. But for my other 3 tires, they look like they still have a few thousand miles of thread wear left in them, so I'm hesitant to replace them with new tires. Is there any danger if I run this setup where one of my tires has about twice more thread than the other tires?

 

You should always try to replace all four tires at the same time, however if that's not possible than at least try to replace two tires and put them in the front.

 

I'd suggest getting that one new tire shaved down to the same depth as the other tires..
Yes, its hard to remove miles from a brand-new tire, but realize that with different height tires, the differentials will be working as if the car were in a steady state turn for however many miles the diffs last until they puke due to excessive wear.

Its possible warranty work on the drivetrain could be denied based on running tires of different diameters