Aloha!
I get the distinct impression that there are not a lot of iaidoka on this thread. My wife & I hold dan ranking in Muso Jikiden Eishin-Ryu Iaido, so let me spend a few moments & provide some
accurate information on Japanese blades.
First, the katana is more a length of weapon, rather than the weapon itself. The long sword, called the daito or tachi, is between 24" & 48" long; the shorter version of these is often mistakenly called katana. The next shorter weapon is the wakizashi or shoto, which is between 12" & 24" in length. The shortest weapon is the tanto, & is shorter than 12". A set of swords with the daito & shoto together is called daisho.
Next, iaido students (iaidoka deshi) usually begin studying using an iaito. The iaito has an unsharpenable blade, usually made of zinc-aluminum or zinc-beryllium alloy. I have carried an iaito in Japan & six other countries with no problem whatsoever -- but I was smart enough to carry it in its protective silk cloth covering inside a sword bag, which is why there were no problems. The sword itself in Japan is highly revered, & if you don't treat it that way, your Sensei will quickly kick your butt out of the dojo, whether or not you practice in Japan.
The bokken, or bokuto, is a wooden weapon that is used in several practice modes. It is the same size & general shape as the iaito, & weighs about the same, but is seldom used except for learning & practicing kendo kata. These kata help deshi, especially kohai (new students), to learn important factors like ma-ai (distance between opponents) & timing of their swings.
The shinken, or "live blade," is also used in iaido for advanced practice as well as for tameshigiri, or practice cutting of tatami omote & bamboo. Your Sensei will likely not allow you to practice with shinken until you have reached dan rank, & maybe not even then, depending on how consistent you are in practice with nukitsuke (the initial draw of the blade) & noto (resheathing the blade after cutting). The shinken blade is razor-sharp, & unless you prefer picking up your body parts, listening to your Sensei is an excellent idea. You must also specifically advise your Sensei any time you bring a shinken into the dojo for obvious safety reasons.
As for wearing gi & hakama on the street, I have some mixed news. Unlike dojos in the U.S., when you don these garments in a Japanese dojo, you are expected to strip naked, indicating your willingness to hide nothing from your Sensei. This is no big thing to the Japanese, of course, with their communal public baths, but many foreign iaidoka prefer to wear their gi & hakama to the dojo, which means wearing them on the street. I never had any negative reaction in Tokyo or Osaka, but was questioned once in
Kyoto when I walked by a koban (police kiosk); they were extremely polite when I explained where I was going.
As a long-time sword collector & Japanese Sword Society (JSS) member, I have also learned a tiny bit about collecting Nihontô, which is the correct term for non-practice bladed Japanese weapons. First & foremost, because the sword
is revered in Japan, there are most definitely special rules & regulations there that apply to all blades, & especially Nihontô. According to a JSS study, fewer than one Japanese in 100 has ever studied a martial art, & of those less than one in 100 has ever handled Nihontô. So you shouldn't be at all surprised when the man on the street reacts strongly to the sight of a blade.
Starting with the most "precious" weapons, Nihontô holding special origami (papers) issued by the Nihon Bijutsu Token Hozon Kai (NBTHK) are the most closely controlled. These blades have been judged from the "rank" of Hozon (Worthy of Preservation) to Tokubetsu Juyo (Extraordinarily Important Work) by this government organization, & can only be imported or exported from or to Japan by licensed specialists called shinsa. An unpapered Nihontô can sell from a few hundred to a few thousand dollars, while a Tokubetsu Juyo blade seldom sells for under $100,000. There are many advanced collectors who travel to Japan to try & uncover Nihontô which they export back to the U.S., & then sell to us less-traveled collectors. They require special licensing to export blades.
Iaito & bokken can be easily shipped back & forth to, from, & within Japan, as long as they are carefully labeled as "sports tools." However, it is currently illegal to bring a shinken
into Japan, although they can be exported without any special paperwork. Similarly, wearing a blade on the street in Japan is illegal unless you have license to do so as part of a cultural arts group; I'm not aware of any license that is available to an individual, although there may be exceptions. There may be a way for an individual to bring in Nihontô to Japan, but I would play it safe & pay a shinsa to handle that for you. And by Nihontô I include gunto (World War I & II blades) as well as older weapons.
Hope this clarifies a few things.