Kevlar tyres. Or at least kevlar liners. Check the pressure regularly, especially if running on patched tubes.
I suspect sharp debris/litter may be the problem
Lining with old tubes doesn't seem like a good idea - they're no stronger than the tube you've got, and add significant bulk affecting how the tube sits.
For properly bad roads, I wouldn't ride on road tyres - get something designed for rougher urban use. All the major tyre manufacturers make something with kevlar, slightly thicker rubber and a bit of grip compared to road tyres. I've fitted continental's version to a mountain bike for city use, and it will take much more sharp stuff than MTB tyres.