I can't really help with the first couple of questions as we went in to Tokyo itself first and on to Disney a few days later. We did visit IKSPIARI one evening whilst were at Disney - it is essentially like a huge shopping centre, so you can almost certainly kill time there but it may not be the most interesting.We didn't stay at the Hilton (we managed to book Disney's Ambassador at a good rate so stayed there instead!).
We bought our tickets when we arrived, which feels odd when you are used to doing everything in advance. I think the prices are pretty set. We paid a small amount extra to have park hopping after the first two days, but I seem to recall that there was no option to hop on the first two days and you had to nominate which park you would be in.
Why do you consider Airbnb a no-go for next year? There has been a change in the law which has placed restrictions on where and who can offer rooms via Airbnb, and formalised the registration process. This caused short term chaos earlier this year, but anything which is properly licensed in an allowed area is fine. Airbnb - and the Japanese authorities it seems - are very "on" this, and won't be allowing listings that don't meet the regulations, so I'd still happily book with them.
Uber operate in Japan, so are a potential option. We had planned to used them for our transfer from Tokyo to Disney, but unfortunately Uber went down that morning! I used instead a very similar service called MyJapanTaxi which was excellent (although the confirmation email was entirely in Japanese, so I needed to make use of translation!) The cost seemed reasonable to me (sorry, cannot recall it) but I was happy to pay a bit for an easy life. On returning from Disney to Tokyo Station, we just had the hotel call us a taxi. This was definitely slightly more expensive than the MyJapanTaxi service.
Can't help with the seaside either, as not been there. However with 4 nights you have plenty of time to get to somewhere like Kyoto, which is my real must-do. It's a fantastic city and a real contrast to Tokyo.
Train tickets - broadly, yes, you can buy as you go, however there are some tourist options that may be worth thinking about. For example Tokyo Metro have tourist passes available only to foreign visitors
tokyometro.jp/lang_en/ticket/value/travel/
However, you need to have some sort idea of where you might want to travel, because unlike in the UK, you need different tickets to travel with different operators. So the pass I've linked to is only valid on Metro and Toei lines, not JR lines.
Based on the itinerary you have here, a JR Pass probably isn't going to be worthwhile, but if you do look in to travelling around then you need to start adding costs up to see if it will save you money.