As mentioned before, one of the best ways of 'taking in the culture' would be to factor in a Matsuri (festival) into your trip. There's a link in my last post about the Omihachiman Shrine fire festival, but you haven't even mentioned if this is something that interests you.

Apart from showcase events (like CEATEC) and at showrooms Japan isn't so obviously a cutting age computer driven society as you might imagine. No to say evidence doesn't exist, it's just not obvious... you'll need to do some digging up. I find it hard to keep up with where this technology is displayed in Japan (and I have a tendency to focus on the Arts side of technology). For instance, you will just have missed "The Great Robot Exhibition: Karakuri, Anime and the Latest Robots" which ended a couple of weeks ago at the National Museum of Nature and Science. Here's a blog showing other robot stuff going on in Japan http://www.lovingthemachine.com/labels/robot.html
I'd suggest stopping at Nagoya to see the recently opened Robot Museum there http://www.robot-museum.net/ (edit - nope! see post #41)
In Tokyo for technology driven Arts I like the ICC in West Shinjuku (near the tokyo metropolitan government building)
http://www.ntticc.or.jp/index_e.html
If this is what you all what to unearth and explore I recommend that you open a new thread here asking specifically on technology/computer focused places, events and tours.
As to where you'll want to go out, throw a stone. Tokyo is teaming with bars and izakayas (where you eat as well as drink). If you're aiming to meet other English speaking tourists and longer staying visitors, as well as Japanese people who also seek out this kind of company, then, as you've already suspected, Roppongi will be a draw for you.
If you want to go to a club then the suggestion I always give is to go to one of Tokyo's superb independent record stores and ask about events that reflect your own tastes in music. Once you go to one and speak to people there it's easy to find about others If you are visibly enjoying yourselves in a venue where you like the music and feel comfortable and at ease you are going to be a much more attractive prospect than a group of tourists huddled together and looking nervous and bemused enduring music they'd never listen to 'at home' (Gas Panic might have a few of these, especially during their Happy Hour).
A word of caution about hotel room policies. In most places, single rooms are single rooms. The same room can be double the price based on two person occupancy rather than one person occupancy. Don't assume you'll be able to sneak someone back.
If you've found economic accommodation, it might make more sense (be less embarrassing) to just buy a stay at a Boutique/Love Hotel for the night than negotiate an additional person into your room. Getting rooms on points at hotel chains is often a great way around this. Alas, the cheaper the hotel, the more tricky this issue becomes, and with expensive joints it's probably best to negotiate with reception first in case there's a two person supplement. (On both ends of the spectrum the front desk tend to be extremely observant).
You also haven't mentioned whether you are aware of the current Tsukiji Fish Market discussions on this forum about Tsukiji policies.
Here's a newly opened thread: http://www.flyertalk.com/forum/showthread.php?t=790617
as msb0b has said, not a good time to go hiking up Mount Fuji. People die up there in March.
Cheap sushi (IMO) is a waste of time. Sushi is easy to find in so many cities now, so conveyor belt sushi isn't going to be much different than what you'll get for a moderate price in Toronto or Vancouver. I suggest spending less on cheap/average sushi and splurge instead on great sushi at least once. Outside Tsukiji market is a good place to find some. Search this forum for more suggestions.
Concentrate instead on trying foods you aren't so familiar with. Chances are (in fact chances are extremely high) that you won't have experiences great Ramen yet. There can't be more than a handful of places outside Japan where truly GOOD Ramen is served. You may find that this will be a revalatory experience for you once you arrive. Best of all, it's cheap.
Try and hunt out a great Okonomiyaki joint in Osaka (a regional speciality you are probably best off avoiding in Tokyo). And MegatopLover's Osaka steak house suggestion is one I would trust completely.

Nothing suggested so far is expensive, except the sushi. Standards are pretty high elsewhere, so if you want a sushi experience of an 'only in Japan' calibre, you need to pay for it. Thankfully, you do get what you pay for.
For more modest fare, stick to hunting out local specialities, and try sating your hunger for gourmet fare at lunchtimes (Tokyo is the only city where you can have a meal at a Michelin starred restaurant for around the $20 mark)
Here's a thread http://flyertalk.com/forum/showthread.php?t=468207

It completely depends on how interested you are in shrines. It's not a question that makes sense for those touring/visiting Europe. If you go to Milan you'd probably want to see the Duomo, Paris visitors are drawn to Sacre Cour and Notre Dame, in London people want to see St Pauls and Westminster Abbey. In each city there are many more churches for anyone with an interest in their architecture, the Art they contain, their history, their events, their denomination.
Japan is the same. The most celebrated shrine in Tokyo is the one in Asakusa, the shops around it are as big a pull for tourists as the shrine itself. Other than that, Tokyo has hundreds of shrines and temples of enormous interest for those with an interest. If you don't have that interest, they may hold little attraction for you.
Osaka probably has less of interest.
That this is even a question for you, my suggestion is to spend a day in Nara (near Osaka) and depending on how you enjoy that, then spend some time exploring the temples in Kyoto. A lot of people get 'temple fatigue' in the kansai area, I suspect your group are potential victims to this.

Already mentioned a festival. Almost everything else mentioned has been unique too. You're also visiting at the wrong season to see lots of traditional attire (outside the festival and, possibly, the cherry blossom viewing events). If you keep your eyes open you'll see people wearing kimono everywhere. It's in the warmer months where you'll see more people donning traditional attire.
Kimonos are expensive, difficult and time consuming to put on and a bit constricting to wear (kind of like wearing corsets - if someone at the wedding you are going to is wearing one, be sure to pay a compliment... it will go down well, I promise). Not everyone enjoys wearing them. There is another version of traditional attire called a yukata. Many visitors think they are the same thing, but yukata are made of cotton and are much, much easier to put on and wear. There's been a bit of a boom for these recently.
If you've seen photos of ladies looking relaxed and casual in kimono with fans in their hands, chances are they are wearing yukata. This means you are more likely to see more people in traditional dress between May and September.