Repair or replace old copper valves?

I'm having the strangest problem with my kitchen sink ... one that seems to defy the laws of physics! My pipes are clogged. Normally, I can take the pipes apart by unscrewing them and simply scooping out the crud in the pipes with a spoon. But this time, the clog seems to be so far down that I can't reach it. I've tried liquid clog remover, but that doesn't work. But here's the weirdest part: If I put water in my sink, it will take about 10 minutes to go down the drain. That's slow, but at least it gets the job done. But when I took the pipes apart to attempt (in vein) to scoop out the crud, I noticed that the water in the pipes just doesn't want to go down. At all. I can leave it there overnight, and the water in the bottom pipe (the only pipe I can't unscrew) is still there! Now that's the part I don't get! If the clog is so strong that the water doesn't drain at all, then how can the top water still drain? If anything, shouldn't the weight of the top water press down on the bottom water and push it through? After all, that's how water towers work! [url=http://mentalfloss.com/article/64577/how-do-water-towers-work]How Do Water Towers Work? | Mental Floss[/url] But the water in the top pipe still drains ... slowly, but still drains ... while the bottom pipe is totally clogged! If anything, shouldn't it be the other way aro I'm about to call my landlord for help. But this is weird! Does anyone have an explanation as to how this could possibly be happening?

The drain for the tub is made of a black plastic material and the tailpiece doesn't have threads. What fitting is required to attach the tailpiece to the 1-1/2 dia. pvc pipe? The tailpiece, also 1-1/2 dia. will fit snugly inside the pvc pipe. Thanks Larry