I recently spent 5 days in Tokyo, we stayed at the Conrad located at Shiodome so very easy to use public transport. Will filled each day exploring major areas of the city (plus a day at Disney Sea) and partying in the night either until the last train or the first train in the morning!!

After 5 days we came away with still a list of places/things we want to do in the city, plus a definate urge to explore more of Japan outside tokyo.

To address your questions
1) I would definately contact the hotel in advance and advise of your early arrival in case they are able to allow you an earlier check in, if this is unlikely and you dont want to cross town to drop bags, then at most stations there are a lot of luggage lockers so you could always head somewhere and leave your bags locked up while making the most of your day. Or even with your early hour arrival try and get to the Tsukiji Market for some sushi breakfast (you likely be too late to snag a limited spot at the tuna auction)

2) We didn't leave the city in the 5 days there for this very reason as we didn't want to spend valuable time travelling for hours, although it is possible to do a day trip out towards Mount Fuji / 5 lakes area.

3) this is all down to personal preference, do you want to shop, see temples, parks?? Some good areas to look at are Shibuya, Harajuku, Ueno Park and Markets, Asakusa, Imperial Palace Gardens, Ginza, Roppongi, Tokyo Bay off the top of my head.

If your going to use the train system within the city both overground and underground then get a PASMO card which works in similar way to the oyster card in london which you charge up with cash. The machines at all stations will have an english button so it's not as difficult as it may seem.