How much can a 700x23 inner tube stretch? I recycled my dad's road bike (about 20 years left in storage) so I bought a complete set of new wheels. "Good" hubs and rims, not quality brand though but I've been told they're good and that's what almost everyone in this country uses. Now the thing is that the first time I inflated the front wheel after a 1km ride it exploded due to over-inflation. After that I was so careful not to over-inflate that I under-inflated the rear one and got pinch flats. While trying to figure out what pressure I should use I came to realize that the tires are 28" x 1" 1/8 (700x28) and the inner tube is 700 x 18-23 which to me seems quite a lot. I should note that I weight about 85kg (something like 180 pounds I guess) and as I've been reading for that weight/tube size I should be putting about 116 psi and the tire is rated at 85 psi max.

I don't care much about pressure since I do not intend to compete, just commuting and eventual light-training in the city. To sum up: am I getting flats because of smaller sized tubes? Should I get bigger ones (or smaller tires)?
Any help is highly regarded!
Yes the tube size should match the tire size.   But a smaller tube will stretch.   Too big a tube may fold.

Max means max.  If the the tire says 85 psi max then that is the max. 
If you weigh 180 then run the tire at or near the maximum.

Do not put 116 PSI in that tire.  Even if you have the wrong size tube pressure is still based on tire (not tube).