Space is big. Really big. You just won't believe how vastly, hugely,
mind-bogglingly big it is. I mean, you may think it's a long way down
the road to the chemist's, but that's just peanuts to space.
--Douglas Adams, The Hitchhiker's Guide to the Galaxy
The distances involved are literally astronomical.
A "high-density area" in an asteroid belt is still going to be better than 99% empty space. Get the movie images of ships being immediately pelted by thousands of rocks the moment they cross the "boundary" into an asteroid field out of your head -- that's complete fiction.
Case in point: We've sent numerous probes through the asteroid belt to regions beyond (e.g Voyager I, which has "left the solar system" a couple dozen times...), and did absolutely nothing to evade asteroids or protect against them while passing through the belt.
That said, you put a big, "stationary" industrial processing center anywhere in space, and you do run the risk of asteroid impacts. So figuring out ways to protect your center is still a good idea. Most asteroids are essentially pebbles, really, so just using a tough outer shell will cover you against greater than 99% of the impacts you'll encounter during the lifetime of the facility. [Not to imply that dust/pebbles are harmless, mind you, just that, relative to the "big 'uns", they're easier to defend against by just using a really tough outer shell; in any case, it's barely a marginally-increased threat in the asteroid belt versus anywhere else in space.] The really big rocks are so few and so far between (literally astronomical distances between them) that your odds of ever being struck by one really aren't worth considering. If you really want to be paranoid, though, you could always find a big asteroid and mine into it, then build your processing center inside.
And if in the off chance a big asteroid is heading straight toward you, you'll generally have plenty of time to put together a crack team of aging ex-astronauts to fly out to the thing, drill a couple of holes, and set the nukes. Or just get out of the way. Or, if in the unlikely event neither is possible, just evacuate the facility, consider it a loss (if it actually does get hit) due to natural disaster, collect the insurance payout (you did buy the asteroid rider on your insurance policy, right?), and then rebuild (just like terrestrial industrial facilities have to be rebuilt after hurricanes, tornadoes, earthquakes...).
Really, working in the asteroid belt is no more dangerous than being in any other part of space. Maybe a 0.000000000001% increase in your odds of getting hit by a big space rock. Given all of our history of space flight, going to space is far safer than crab fishing in the Bering Sea, and working in the asteroid belt is very unlikely to dramatically alter that fact.
And ultimately, no matter where you put it, mining ships will have to catch up or fall back to your facility. That's just how space travel works (here again science fiction movies have lied to you). Everything's in its own orbit, and rendezvousing always requires moving to higher and/or lower orbits. The only exception would be if you put your processing facility directly on or in orbit of the rock you're actively mining, but at that point it might make more sense to just incorporate the processing facilities directly into the mining ships themselves so they can always take them with them!
And the waste? Just chuck it into space. See again: Space is very big. Kessler syndrome is exceedingly unlikely to cause you any problems out there -- it's really only a (potential) issue in low-Earth orbit, and that's only because we're talking about a very small amount of space. If you're really worried about it, pulverize it into dust first. You're probably doing about the same thing to extract the ores you're actually after anyway.