Gotchya....so when you cycle a brand new tank, what is it that takes the nitrates back to 0?
Water changes.
Normal process is to cycle the tank with nothing living in it. At the end of the cycle, when ammonia and nitrites have spiked and dropped back to zero, you'll have some amount of nitrates in the water. Since there isn't anything living in the tank, you do very large (like 50% to 75%) water changes... not worrying about matching salinity, temp, pH... anything. You keep doing this until your nitrate levels are where you want to start things with, because you'll probably never get them this low again! So preferably, you do water changes until you can't test for anymore nitrates.
Nitrates are normally considered the last step in the nitrification process. Ammonia turns to nitrites, and nitrites turn to nitrates. And then you do water changes to gete rid of the nitrates. If you have oxygen-deprived areas of your tank that harbor yet another type of bacteria, then you can convert the nitrates to nitrogen gas and it will bubble out of your tank. This normally occurs in "deep sand beds", or in other types of setups that use special media to take the nitrates out of the water. Probably beyond what you're wanting to deal with now, but file it away to look into for that new tank.
Bioload, as we're talking here, is pretty much always referring to just fish - things that will create "food" (ammonia) for your beneficial bacteria. Coral doesn't mess with your beneficial bacteria one way or another. Once you get to a certain quantity of coral, it will start to mess with your water parameters though - specifically your calcium, alkalinity, and magnesium levels. It will draw those things (and others) out of the water to grow, and those things need to be replenished. That's either done with those same water changes you're doing for the nitrates, or by adding supplements ("2-part solutions") to keep those levels constant. Depending on your coral stocking levels, and the brand of salt you use, normal water changes can often keep up with the demand from the coral. Another good reason for water changes.
Sorry... more info that you need right now, but figured I'd get it out there so you have more info to digest!
Regarding the fish gasping... not sure if nitrates at 80 is enough to really cause breathing problems, but I've never experienced high nitrates so I don't have any first hand info. Like thincat mentioned, I wouldn't recommend massive water changes. I'm sure you've heard the saying by now, something to the effect of "Nothing good happens fast in a saltwater tank.", meaning be patient and don't expect results overnight. I've also looked at that saying to mean don't make rash decisions or do anything quick in a saltwater tank. There are very few things that need immediate action.
You can safely do 25% water changes every other day, which will take that 80ppm nitrates down to under 20ppm in 4 water changes. You could probably do that 25% every day - as long as you match the temperature and salinity pretty well.
[Edit: Just read that article you linked to. I definitely would NOT try this. One little bit that they casually gloss over is that part where they talk about adjusting the pH in the old water to match the new incoming water. What they don't mention is that this pH adjustment process in itself, can really screw things up. It is really hard to raise pH and keep it there without really messing things up. They also compare nitrate in fish to carbon monoxide in humans. Just my opinion, but from my reading, this is not a fair comparison. Nitrates, at levels "normally" found in our tanks, are not toxic to fish. It's better if it's not there, but it's not going to kill them outright like CO can with humans. If, as they mention in the article, you've let your tank go unmaintained for that long that your nitrates are out of control, then what does it hurt to slowly lower them over the course of a couple weeks as you do smaller water changes? Yes... it's inefficient - but it's way way safer.]
