Closure is always nice, so here it is.
Okay, my son and I managed to replace his front speakers without dropping the dashboard or drilling out any studs. Here's how it went.
1. First of all, the Polk speakers that are supposed to fit right in don't fit right in. They have a multitude of holes of varying diameters in various locations, and none of them fit. The closest fit was some notches in the outside of the plastic frame on the sides. They were just a little too close together in the horizontal direction, and very much too close together in the vertical direction. So the first thing we had to do with the beautiful new speakers was goober them up with a coping saw. Not fun.
2. On the passenger side, by removing the interior light below the speaker, there was enough room to get a hand up and find the studs. The nuts were too tight to loosen by hand, but all we had to do to break them loose was to grab a 11/32" 1/4" drive socket and use it to turn the nuts. Just a little extra diameter is all you need to break them loose. Then we spun them off by hand. After backing the speaker off the stud, with a little contorting, we were able to lower it out of the car. The new speaker went in the same way. We used fender washers and attached the speaker using three of the studs. It was tight enough.
3. On the driver's side, we removed the emergency brake pedal by removing two nuts under the hood and one under the dash. We also removed the interior light, and there was a kind of little "skid plate" that appeared to be there to protect some wiring. We removed that. Then we removed a bunch of torx-head screws holding the left side of the dashboard on, and pried it back and stuck a small piece of 1/4" aluminum plate that we had to keep it pulled out. Even after that, access was more difficult than on the passenger side because of the plate under the dash that provided a mounting point for the emergency brake pedal. In spite of that, there was still enough room to get a hand up there and remove the nuts holding the speaker in place. Getting the speaker out required two people: one to pull a large wiring harness out of the way, and the other to raise the speaker in place, which required a lot of turning the speaker left, right, up, down, in, out, etc. Again, we attached the speaker to three studs using fender washers.
The steps I described above were not done efficiently in the order I listed them. For example, we didn't discover that the speakers had the holes in the wrong place until we had been trying for almost an hour to get the passenger side speaker into place. Then, depressed, we walked away from the job for almost a week, during which my son had no music in his car, which sucks.
So to conclude, we didn't need to drill out any studs, and we didn't need to lower the windshield and remove the dash. However, it would obviously be a lot easier to do this if you first remove the dash.
Thanks a bunch for everyone's comments, and I hope my writeup helps somebody in the future.