It does sound like a tandem is a good option for you and your wife. A trike for her (many recumbents are trikes) might solve the balance problem, but not the speed issue.
There are definitely folding tandems out there. Might even be able to fold up and stick in the back seat.
The "simplest" way to get what you want is an S&S Coupled bike with a single split in front. That's a bike that doesn't "fold" but splits into two pieces (or 3). If it splits in front, you should be able to put it on a car bike rack and won't have to mess with taking a chain off. It's possible to retrofit many non-aluminum bikes into a folding/travel bike this way.
I believe usually people get an S&S coupled tandem so they can fly with the tandem and not pay outrageous shipping costs. Since you're more interested in getting a bike onto a bike rack on the back of your car, you should have an easier time finding options.
So, the good news: yes, options exist. The bad news: they're custom or semi-custom and generally expensive.
Take into account that tandems can be hard to fit properly unless you're the "right" size difference.
Your best bet is probably locating a couple framebuilders near you and talking to them about your needs. But look around at a few of these links to see what some options are:
Folding Tandem from Bike Friday folds up small enough to fit in back seat. Custom. The "example" build prices that look like a fast road bike tandem setups seem to start around $3000 USD.
Zinn Cycles Tandems: I think the Titanium S&S coupled tandem is a model I saw at a big group ride. "$12,000 and up", but the steel travel tandem could be as little as $8000.
Calfee seems to have carbon road tandems with S&S couplings available for $8000.
S&S Coupling US Framebuilder List: Look on their site through the photo gallery and retailer list, too. And at the tandem page. Oh, and the tandem retrofit photo gallery.
(If I could afford it and had those needs, I'd talk to my local framebuilder since he's on the S&S framebuilder list and does a bit of tandem stuff even if he doesn't list prices for them yet. But framebuilders closer to you are probably a better option than one in my town.)