Okay! So thank you for being patient.

Let's start by saying I am not very familiar with the Zatoichi series but I can comment on the various things you mention here! Unfortunately this means that I cannot adequately answer your first question. Let's move on to number 2!

2. Zatoichi is often mistaken/employed as a masseuse. Why was this profession common amongst the blind?

Yes, being a masseuse was indeed a common profession for blind people! In fact, by the middle of the Edo Period, it was legally only allowed to be practiced by blind people!! It was through this that massages became heavily associated with the blind.

Another fun side fact is that there was one other profession reserved solely for the blind which was the recitation of epics and songs, usually for blind women. Blind people who took part in these professions were highly organized and were formally under the authority of a trades guild for these professions. The men at the top of this guild were extraordinarily wealthy!

3. In one film, Zatoichi stumbles across a village that doesn't allow gambling, fighting, or whoring, and comments that this is unusual. Would it be unusual for a village of the time to be missing one or all of these elements?

Gambling, whoring, and fighting were generally illegal across all of Japan except under extremely heavily regulated instances in certain areas. Subsequently, these areas became famous (or notorious) for legally sanctioning these activities (red light districts in the capital, gambling areas in Nagasaki, etc)

That being said, private gambling was probably allowed to a certain extent for many of the same reasons the police don't knock down doors to raid your average poker game. It's very hard to enforce and not worth the time and effort to do so.

Criminal gangs that ran gambling dens or unsanctioned prostitution rings were generally investigated and sought after by government officials.

Your average small time village in Japan probably did not have the population or pool of money to make it worth setting up an actual establishment that provided gambling, prostitutes or fights. It's important to remember that Japanese villages were generally very currency poor because farmers paid their taxes to lords directly as grain. Many villages lacked much in the way of coins and often bartered when trading.

That being said, small time/independent prostitutes, private gambling, or your random fist fights were probably still fairly common. Unless they became a huge social issue, villages probably didn't bother trying to enforce no tolerance policies.

4. There are a number of times that Zatoichi stumbles across archery booths, similar to what you'd see in a modern county fair. Did these actually exist? Were they parts of traveling carivals, or permanent fixtures?

I have never personally heard of this or come across examples of this. During this time period, and stretching back for hundreds of years prior, the bow and arrow were the symbol of the samurai. Horseback archery was the number one way that a samurai showed off his martial skill and discipline.

Even without that context, a bow and arrow were still considered military arms and the local peasantry would not have been allowed to randomly brandish weapons for no reason. The Tokugawa Shogunate was extremely strict in enforcing arms bans across Japan to consolidate their power and monopoly on arms and violence. So following this line of logic, it would probably preclude the existence of archery booths. At the very least, I would imagine they were highly unusual and definitely not permanent.

5. If you did find a dice game, what were the chances that it was using weighted dice or otherwise rigged?

Talk about loaded dice comes up often in Japanese literature. Dice games are also still being run today by criminal gangs in Japan and the idea of losing because of loaded dice is very common. But at the same time, groups that run gambling establishments need their patrons to keep coming and that generally means having quasi-fair games. Usually gambling game rules are already made in such a way that the house already has a slight advantage.

Another game that was popular for gambling in Edo Period Japan was hanafuda, literally 'flower cards'.

This is another gambling game where the house cheating or rigging the game is a common trope in Japanese culture.

But like with dice games, in most gambling situations, the house rules already gave them a natural slight advantage which meant that on numbers alone, they would come out on top over time. Same as modern day casinos. No need to do something risky like using loaded dice or switching cards out in decks.

6. Zatoichi often identifies himself as a Yakuza, despite the fact that he doesn't work for any organization. What does Yakuza mean in this sense?

This is almost certainly an apocryphal use of the term since organized criminal gangs would have never called themselves yakuza or even know what the term means in the Edo Period.

7. Plot aside, which depiction of the Edo period in media most accurately captures the essense of that time period?

I think Kurosawa's film Red Beard is an excellent down to earth depiction of Edo Period Japan that isn't about samurai/ninja/ronin or other almost mythological characters. It's simply about life, the status of the social order in late Edo Japan, and almost mundane day to day existence. I think it captures a small aspect of Edo Japan that's much harder to really get a feel for in other action oriented movies/books/anime/etc where the focus is on exciting plot developments and not life in Japan itself.

I can only speak to 5; gambling was officially illegal but the prohibitions were not heavily enforced. Most dice games would have been run by the yakuza, so they could certainly have loaded their dice. However, there's a biography of a yakuza where the narrator says something to the effect of "it's a bad idea to load dice, because if you're clearly cheating customers they'll go gamble somewhere else."

I've certainly heard of other blind masseuses in East Asia, so my guess is that it would be common, but I honestly don't know for sure. Can't speak to the rest.