So here's my update on the cleaning project so far. (And just to be clear, none of this track has been laid yet - it's all sitting on a workbench in my shop). I've been using a small amount of mineral spirits with gray Scotchbrite as mentioned in the original post, and it's been working great. It's taking off much of the original glue and bits of ballast left behind by the previous owner, and not damaging the ties or the rail. It's cleaning up the rails just fine. For the rest of the gunk on the ties, and to clean the ends of the rails for the connectors, I bought a pack of brass (not steel) wire wheels on eBay - super cheap, 45 piece set for 11 bucks - to use in my rotary tool. I have a Foredom - like a Dremel on steroids. Came with three different configurations of wheels, so they've been great for getting into all the nooks and crannies without damaging the track. Not a terrible amount of effort, and the track is coming out great.
Thanks to everyone who responded with suggestions!