Looks like the handiwork of a leaky windshield.
Water leaks in around the windshield, runs down the inside of the firewall, and pools on the floor under the carpet. The floor gets rusted out from the inside, and no amount of undercoating on the outside will stop that. Sealing up that windshield is the solution.
As for the adhesive, go for it. Some new cars are completely assembled with it, so replacing a section won't be a big deal. Just be sure to leave good amounts of overlapping metal for the adhesive to bond to. You'll probably have to screw the panel in place while applying the adhesive, then after it has cured enough, grind the screws off.