Well, let's see if I can answer your questions, with the little experience I have. Learning what is proper etiquette in Japan is a non-ending journey. My first 2 years in Japan I tried to learn it all, and in the end I just said, "screw it" and realized that more people liked me when I just acted like a "barbaric"
American. LOL! So, many people have been in your shoes before and will be after. If you are just going for a few practices you can just pay a mat fee. I do not think you need to show papers or anything like that, but do bring your affiliation card with you if you have one. If not I doubt it will matter, I doubt they will turn you away, as long as you are not that of a black-belt rank the politics are not that big of a deal. But you are a foreigner, so you can get away with a lot of "etiquette" mistakes. The funny thing is if you do something wrong the only people that will say anything will be other foreigners. So just ignore them. LOL!
I am sure the Japanese people there will be patient with you and want to practice their English on you. But yes as Donna said, they do not speak much English at the front desk, and if they do they are too shy to use it. But just think of it as all part of your "Japanese Experience." As for classes, you can attend any of them. Do not be afraid. True there is proper etiquette to attend a class, but again you are not a native Japanese person so do not worry about it. Nobody will tell you off. After all Hombu Dojo is a place to promote Aikido to the world so they welcome all. Going to smaller dojos are a little different, but again you are a tourist so just let them know ahead of time you are coming, but if you do not, no worries, I am sure they will still love to have you. Smaller dojos are great, to me that is where you will get your best Aikido experience. Small Japanese dojos really have a family like quality to them. When you are at Hombu if you meet a red head English guy name Brain, say, "Hi" for me.
Anyway, I am not an expert on this matter by any means, but I know what you are going through I was in your shoes once. Have fun.