I'll need to be brief and I'm typing one handed
The bike tours aren't of central Tokyo. Save it for a future visit.
I'd recommend you kept your options open with the Limousine Bus. They are more regular to some hotels than others. It could be that you'll clear immigration at an inconvenient time, rather than wait 90 minutes or so for another bus, give yourself the option of getting a train (the NEX/Suica deal is a good alternative).
I don't think a prepaid inclusive 'pass' (like the JR pass) makes sense, but a preloaded prepaid train ticket (like SUICA or PASMO) does.
Shinjuku should be fine as a base, particularly if you want to be up and about late at night. Don't feel you need to stick to it though should you find a hotel deal elsewhere.
Mt Fuji, day trip, early June? Sounds miserable to me.
Kamakura, Takao, Hakone or Nikko would be 'better'. But for four days I'd leave the day trip option as a standby - you won't even make a scratch on Tokyo in that time. Try changing the pace of your visit by exploring the Yanasen area - the downtown districts of Yanaka, Nezu and Sendagi - or/and Kagurazaka.
There are cheap and easy to use laundromats all over Tokyo. Your concierge can help you locate one, many can be found at the entrance of public baths. The cheaper hotels often have this option, with more expensive hotels you'll pay per item at a huge premium.
Take cutlery with you and go anywhere you like rather than feel compelled to stick to Italian or Yoshouku restaurants. Or you could invest in the adult version of these:
http://trainingchopsticks.com/ and go native
Tokyo is full of economical options, you'll be spoiled for choice.