Hey Matt - welcome to the forum!
To be honest I'm a bit suspicious of the birdcage you linked to, mostly because it's painted. That, combined with the fact that it doesn't sit flat on the ground, worries me a bit. Also, they add says it was completely media blasted before painting. It wasn't. There's obviously remnants of old panel adhesive on the birdcage. If they couldn't bother to fully media blast the adhesive off it causes me to question the quality of their work elsewhere. The paint is also very thin and uneven. This seems to be indicative of poor quality workmanship in my opinion.
Specifically, you have no way of knowing what it looked like before it was painted. Was it all dented from an accident and heavily rusted? Did they simply slap on some Bondo and a few patch panels to make it look pretty? Chances are the guy is being honest and wanted to put out a good product but I've been trained to assume all e-Bay ads are scams until I can convince myself their legitimate. I've heard way too many horror stories of people being scammed.
To my mind there are two issues specific to the misalignment issue. First, you'll need to mate the birdcage with the frame. In that area I agree with the seller (assuming the misalignment is modest - a half an inch or so). The frame is much, much stiffer than the birdcage so it'll easily come into alignment when you bolt it to the frame. My bigger concern is why is the birdcage misaligned? I'm guessing the car was in a major wreck. Assuming that's the case how will you know that the birdcage isn't distorted or racked above the level of the frame rails? If it is distorted will the firewall and rear clip fit properly onto the birdcage? I'm having nightmare visions of you trying to straighten the birdcage with come-a-longs and chain falls.
Ultimately, $2k is a lot to pay for a birdcage - you could probably buy an entire donor car for almost that much. To my mind there are a couple of yellow flags that you should really investigate before moving forward. If you decide to move forward with this I strongly suggest getting there to see it in person, inspect it thoroughly with a ball peen hammer and magnet (to test for Bondo patches) and to measure for potential alignment issues. You can also peer inside the birdcage using a flashlight to view the interior for signs of damage or repair.
I realize this probably isn't what you want to hear. However, I wanted to identify the potential challenges as I see them.
With respect to your second question, I'm not familiar with the process of removing the underbody from a C3 but would be curious to hear the experiences of others, too.