In general you are not going to be able to get you Rail Pass until a month or so before you go. The normal process is that what you actually get is a voucher and then when you are in Japan you exchange this voucher for the actual rail pass. The voucher has a finite life time in which it can be exchanged so you can't get it too early. I believe that they have been making changes so that you can/will be able to get a rail pass in Japan without pre-purchasing a voucher before you travel but those changes are since my last trip (April 2016). The process is simple and I normally get my vouchers from jrpass.com who, even if you don't use them, have quite a lot of useful information on their site on rail passes and how to use the Japanese rail system in general.
You can use the pass in two main ways; you can either just turn up, show the pass and sit in unreserved seating or you can reserve tickets using it for free. If doing the latter it's best to have gone onto Hyperdia in advance and worked out exactly which train you want (bullet trains are uniquely named, e.g. Hikari 416) so you can say you want this train on this day from A to B ... makes it a lot easier if the person you are speaking to has not very much English. You can't use the Nozomi trains with the rail pass (the fastest bullet trains) and you cannot use it on non-JR lines. It does mean for instance you can do things like this; I was in Kyoto and decided I really wanted some Hiroshima style Okonomiyaki ... so used my rail pass to get a train to Osaka (about 10mins) and then jumped on a bullet train down to Hiroshima in the unreservered seating. On the way down I saw a castle near the station at one of the stops so when I was on the way back, after spending the day in Hiroshima, I jumped off at that stop, had a wander around te castle for an hour then wandered back to the station and got on the next bullet train towards Osaka ... total cost to me for rail transport on the day ... nothing.
A Suica card (think Oyster card) can be useful in the cities saving you from having to buy tickets every time and can stop you ending up with loads of small change. Whilst different cities have different cards most have been linked so you can use them in the other cities as well. This means my card has been used everywhere from Sapporo to Fukuoka and even with some private rail lines that don't take JR passes will take work with the Suica cards (I've used my on private lines in Nagoya and Enoshima). You can get cards and top them up at most main line stations in Tokyo (and you can top them up elsewhere). There is info on this on the jrpass.com site too. I tend to use the JR pass whilse it is valid and I'm travelling on somethings which takes it and then switch to the Suica card when I'm back in Tokyo and the JR pass has expired. The Suica card doesn't save you money but does make you life easier. You can also use it in some convenience stores in Tokyo, some vending machines and some station lockers.
I'd spend one day in Tokyo as it would allow you to sort out any rail pass things and also if you are staying at the same hotel when you return to Tokyo you could then store some of your luggage there rather than dragging it around (remembering that luggage space on bullets trains is not large at all). This is what we normally do and leaving big suitcases in Tokyo and then only using small ones for travelling around and the hotels we use normally do it for no charge for up to 7 days (or if you sweet talk them you can get a bit longer).
I'm with you on getting direct flights .... at ~11:30hrs for the flight I want to just get on and then off at the final destination. Personally I fly JAL (usually premium economy) normally as I find their ~7pm flight from Heathrow works best for me jetlag wise as, after staying awake on the flight, it gets into Tokyo mid/late afternoon which means I get to my hotel between 5-6pm, shower, go out for some food / top up my Suica card / wander around Shinjuku a bit and then go to bed about 9pm and normally by the next morning I feel normal again. Speaking to someone I know who is Japanese and has flown many different carriers back and forth she says that whilst JAL and BA do code share on flights you can get better luggage allowances on JAL flights than BA ones.
You need to consider as well which Tokyo airport you are flying into, Narita or Haneda ... the former being a lot further out of Tokyo than the latter so taking quite a bit longer and can cost more to get into the city from. Also how you are going to get in from the airport to the city. there are various options for this including rail and limobus. Personally I use the Limobus (orange and white coaches you see all around Tokyo) as it's realy easy, I just walk out of Haneda arrivals and across to the desk and buy a ticket ... then it's only about 50 yards and a lift to get to the Shinjuku stop. From there I can get the coach that loops round Shinjuku and be dropped off directly outside my hotel (in Tokyo I tend to stay at either the Hilton Shinjuku West or the Keio Plaza) ... but there are lots of coaches going to lots of hotels and means you don't have to drag your luggage through crowed train stations after a long flight. If you look around there can be various ticket deals to do with getting the train in from the airports as well if you want to do it that way.
Or what dates were you specifying?