2. There may be hotels with shuttles, but that is over an hour drive, so no "cheap hotels" would offer that service. Really, getting from airport to Tokyo station couldn't be easier, but if you want to wait on activating your rail pass you could look at taking a limousine bus.
Tokyo Station and Nihombashi - Narita Airport | Scheduled Bus Services | Airport Limousine Bus
The limo bus goes directly to several hotels near Tokyo station.
3. I would bring about 1500 in cash and use credit cards at hotels and restaurants where you can.
4. Change your cash into yen before you go. That's one less thing you will have to worry about when you get to Japan.
5. A day trip to Hiroshima with a stop at Himeji is going to be a day with 3+ hours spent on the train. Doable, (and I like riding the bullet train) but just FYI.