Are there road bike tires invulnerable to popping? I am a mountain biker now living in Minneapolis and so I bought a road bike, but I can't keep tires on it to save my life. I 'upgraded' to Continental Attack/Force tires, got a glass shard in the rear, and now just went over a pothole that popped both the front Continental and rear stock tires.

I haven't even been able to even ride a mile on it. At this point, I'll take heavier wheels with more rolling resistance just to be actually able to ride. Does anyone make solid rubber or non-pneumatic tires for 700c wheels?

These look neat, but they aren't being produced yet:


                
            

Side note: I like to drop and hop curbs (though I haven't even been able to attempt it on the road bike), there are frequently metal shavings or other sharp objects on my garage floor, and I live in a neighborhood with lots of cracks and potholes. Should I just give up on road bikes and stick with my mountain bike?
At risk of being downvoted, you don't want want solid tires. There's a reason that all performance tires for all vehicles are pneumatic. The decrease in compliance of a solid foam or rubber tire just doesn't compare to a pneumatic tire.
Look into a tire with kevlar reinforcement- most brands have them- or aftermarket anti-puncture tire liners that sit between the tire casing and your tube. You may even want to try a tire sealant like the one made by Stan's NoTubes, though if you're running tubes you'll need the kind with removable valve cores.
One last thing to note is that the number of flats you're getting may depend on what part of the road you're riding on. If you're riding to the right of the white line (assuming you're in the US), ie on the shoulder then you're going to get a lot more flats. The shoulder is where all the debris accumulates even if the road looks clean. For many road riders, especially those who are just starting and don't feel comfortable riding on the road, simply switching from riding on the shoulder to riding to the outside of the actual lane alleviates a substantial portion of flat issues.