Wow, you get a lot of plumbing questions! I'm not in a hurry, so I can just "bump" this if it gets buried somewhere. No biggie.

I hope I explain this right. For the bathroom sink, there's the (drain) pipe that's sticking out of the wall, the one that hooks to the piece coming off the trap. It only sticks out from the wall a couple of inches. I can't see, but I assume it then is one pipe piece that simply curves downwards.

I was getting slow running way too often and finally took a good look in there. The horizontal part of that pipe was filled up with so much crusted on crap that the actual hole was only about 1/2". (It's a 1-1/4 pipe.) It was very hard and I had to basically chisel it off with a screwdriver.

Well, now then there's the part that heads downwards to the basement. It would be a major job to actually get the pipe out - just trust me on that one. (Yes, even from the basement, which is actually less accessible.)

So I'm wondering if there's some kind of special tool that would get around that corner, or even just something I might have around the house that would work - I've been wracking my brains trying to think of something. Keep in mind, it's crusted on very hard and something like an auger won't cut it. (Not to mention that it's closed off small enough that the auger head won't even go around the corner. So I can't even just clean the regular gunk out.) I need something hard like the screwdriver but that I can use around a corner.

Also, what happens if the stuff I get off just falls into the pipe? Could I potentially be making things worse?

Any ideas? (The pipe is probably iron is that makes any difference at all.)

If that pipe runs to the basement and you have room you might be able to have it removed completely if it is clogged and encrusted that bad. Then install a new pipe. Have you tried an auger? Or have a professional plumber come out and look in the pipe to see how bad it is? They have video snakes that can go inside a pipe to examine it.

Use a 1/4" drum auger snake, it will make the bends. Some drum snakes attach to an electric drill, some are hand crank but both will do the job

I have a drum auger. It won't work for two reasons: one, cause it's so tight in there it won't make the turn. (As in, it literally cannot.) And two, cause that isn't going to clean this hardened on stuff. That'd be a bit like the water slowly washing away the Grand Canyon. I need something that I can put pressure behind and really scrape at it. It took me like two hours to clean the horizontal portion with a screwdriver, let alone a flimsy auger. (And mine can't be hooked to a drill.)

Califdadof3, it can't be reached from the basement without ripping up floorboards. It's up through a tight hole, about 2 feet up. Maybe there's some special tool a plumber would have that can get up in there and unscrew nuts, but I certainly can't do it.

I might be able to get it apart down at the bottom where it hooks to the pipe going into the main drain, but that would mean undoing quite a few pieces, and even then, I can't un-do it up at the top and would need something very long and hard to get up in there and scrape. (And I do mean long, like probably 3 feet or so.)

I can actually unhook everything under the sink and move the vanity, but as long as it's draining (if slowly) I hate to chance screwing stuff up, and that would be some work. I've been trying to avoid doing that, and avoid calling a plumber. (I try to do as much myself as possible.) It tried a coat hanger, which will scrape but can't get any pressure behind it to knock that stuff off.

What worries me and its typical in a situation like this is that the integrity of the pipe could be comprimised by all the banging going on. You could easily cause a fisher or crack to appear. It may very well be that a professional motorized auger is going to be your only option. You can get those from rental places. The other thing is it may have to be done from the rooftop. Where the vent is for the pipe then auger the entire run.

The first question rhat came to mind was, "What's been put down the drain that would harden up like tile grout?" As a rule it's grease and hair that we have to contend with. Can you shed some light on what this hard substance could be? Do you have children? A cleaning lady that empties cleaning stuff down the vanity drain? Help me out here!
As for your blockage,

Morgaine300 I have a drum auger. It won't work for two reasons: one, cause it's so tight in there it won't make the turn. (As in, it literally cannot.) And two, cause that isn't going to clean this hardened on stuff. That'd be a bit like the water slowly washing away the Grand Canyon. I need something that I can put pressure behind and really scrape at it. It took me like two hours to clean the horizontal portion with a screwdriver, let alone a flimsy auger. (And mine can't be hooked to a drill.)

What turn are you referring to? The "J" bend in the trap or the bend in the wall? Did you , in fact, remove the "J" bend and send a anake up into the wall? (see image) Short of replacing the pipes a power auger's your only hope. If I knew what the blockage was that we were dealing with I could be more explicate. Can you gine me any more details? What horizontal pipe so you mean? The trap tail that goes into the wall?
Can you give me a step by step description of what you have tried? Your explanation doesn't cover what I have to know. Back to you, Tom