Floor toilets, sneezing and coughing

I have like many others noticed the frequency of "floor toilets", squat toilets or whatever to call them. Is there any history behind them since they are so usual over here? Are there any particular advantages of using them instead of normal western toilets?

Also I have noticed many japanese people cough or sneeze into the open without covering. Maybe it is just a bad habit by some, so if someone was offended I am sorry. Where I live it is very rude not to cover your mouth if you cough or sneeze suddenly. Just curious.

I have like many others noticed the frequency of "floor toilets", squat toilets or whatever to call them. Is there any history behind them since they are so usual over here?

Yes, they are the traditional style Japanese toilets. Only recently have western style toilets become more and more common in public restrooms. See this page for more info:

http://www.japan-guide.com/e/e2003.html

Are there any particular advantages of using them instead of normal western toilets?

Squat type toilets have several purported benefits. I think one of the biggest is that you don't have to touch anything to use them.

Also I have noticed many japanese people cough or sneeze into the open without covering.

I'm pretty sure that's considered rude in Japan as well. Maybe you just happened to noticed the rude people.

Squat toilets have a long history in many countries around the world, not just Japan.

Squatting over the ground then over a dry hole, that could be filled afterwards with leaves or grass then soil, goes back to the prehistory, well before indoor toilets with plumbing were invented.

Their benefit, according to doctors, is that it makes the "business" easier.

Squat toilets are "common" not just in Japan but in other Oriental countries like China, HK New Territories. I have even seen them in Thailand.
We just came from a 14 day trip in Kanto region and sneezing coughing in the open was pretty rare. I observed more people putting on their surgical masks when they start coughing on the trains.

To use your own terminology, floor toilets are "normal" non-western toilets.

As far as I know, squat toilets were very common in Paris, at least up to the 1990s. They call it "Turkish style."

The Japanese media keeps telling us that squating in a daily basis is very good to keep you healthy in means that it trains your muscles. They even say that nowadays that most of us here in Japan don't squat over the toilet so often, it has affected our health.

Another benefit of these toilets is that it is easier to clean. You can mop or wipe anything on the floor and just drain them into the toilet.

One more great benefit is that, if it's a flushable type, you can clearly see what came out of you, even the amount of it. This makes it easier to check your own health conditions.

Sneezing and coughing without covering is indeed considered rude here in Japan as well. However, if you're not doing it right into someone's face, but facing towards an open space, that's sometimes sort of tolerated as it shouldn't affect anyone else's health.