Some of my input (in Tokyo for 6 years now):
-For a stay of 3 weeks or longer, you can rent a monthly guesthouse room, Sakura house (more expensive, but central, only foreigners), and Oakhouse (lightly less central, but cheaper, and Japanese people are allowed) are well known ones. There isn't really anything cheaper, maybe the internet cafes, but then I would even rather book a dorm at Sakura house for the same or less. The internet cafes are pretty hardcore for 3 weeks I think. The great thing about guesthouses is that you will meet people like you.
I recommend Oakhouse, at a station near the Chuo (train) line. Picking the right train station is more important than location. Check public transport times on google maps, using Shinjuku/Shibya as a reference.
-Trains can be crowded around rush hour, but not so much in between, especially on weekdays. Plan your days so that you don't have to ride the crowded trains, or suck it up.
-Perverse Japanese are not a problem for foreigners. They are too shy and afraid for your reaction, and will focus on Japanese women instead. I cannot give any guarantees though, but your chances are better than in most other countries.
-Yes, August is very humid and sweaty, but there are also many great local festivals (even many inside Tokyo). You WANT to visit these, often the small ones are more romantic than the big ones. Also there will be many mosquitoes.
-There are also a couple of firework shows in august if I am not mistaken, the biggest attracting hundreds of thousands of people. If you are traveling, you could take a spot the day ahead. I don't go anymore though, too crowded.
-Magnetic railways pass (suica/pasmo) is indeed a must. Get it as soon as you can and stop worrying about which ticket to get and enjoy your time better.
-Cool simple day trips are Kamakura (beach, some temples), Enoshima (small Island attached to Tokyo), Mount Hakone (easy hike from Tokyo, beer garden), and Hakone Hot springs. For a 2 day trip Nikko delivers the beautiful Cliche Temple and Nature Japanese scenery, no need to go to Kyoto, which I think is overrated (I like Nara much better).
-Most of these places are quiet during the week, and ridiculously crowded in the weekend. Use your advantage of not having to go to work.
-There are also a bunch of small islands which can be reached from Tokyo overnight by boat (recommended!). If you want to party, go to Niijma in August, and stay at the awesome free camping. you can go surfing there too.
-Go indeed to Shinjuku (west exit) for camera goodness. The big Yodobashi camera and Map Camera are on the same street. I wouldn't bother going to any other store, maybe only Fujiya in Nakano for some 2nd hand digital stuff, they don't have much rangefinder gear anymore. With the high yen you are probably better of looking at the classifieds anyway.
-My favorite area in Tokyo is Kichijouji, also voted as the best place to live in if I remember correctly. Shimo-kitazawa is also nice
-Ginza is boring, Akihabara is boring unless you are a total tech/manga geek. Check out Shibuya crossing, but then quickly run to the much nicer Daikanyama and Nakameguro. Roppongi itself sucks, but has some nice musea. I guess you should see the Mori Museum there.
-Besides sushi, try okonomiyaki, something some foreigners forget, though Japanese people will take you there if you meet them.
-If you go for Korean food, and you should, go to Shin Okubo for the real thing. If you bring western girls they will get treated like princesses.
-Of course you will go to Yoyogi park on sundays, just like everyone else, and that's good. If you come from Harajuku station, also walk towards Shibuya. There is an open stage that often has festivals and cultural fairs in the weekend, which many foreigners completely overlook. The fleamarket there is super fun too. Check here for the schedule:
(google translate it)
-Taking photos is no problem, check my flickr or facebook to see what I get away with. http://www.flickr.com/photos/rudyshots/
-Tokyo doesn't need to be expensive, but it probably will be with all the fun things to do there
I wouldn't eat ricebowls everyday to save money, enjoy and spend, save money at home!