1. I can't speak for every city, or getting _everywhere_ within a city. But I've managed without knowing how to drive, and my impression is that things are much better than the US. I manage with a combination of walking, public transport and taxis. London is especially good for public transport (well, it can still have its problems, but trying to drive through London in a car will just involve lots of traffic and generally won't be quicker). For travelling across the country, the rail network is reasonable, but note it's biased towards London - getting from or to London, or travelling between points on such a line, is well supported, but travelling elsewhere across the country is more of a pain, and in the south may often involve going via London (there's a reason someone came up with the "British Rail" metric space). One downside of not owning a car is that it may limit your choice of where to live - that is, living away from the town/city centre (where prices are of course cheaper/houses are bigger) is less feasible. Another point to note is that overnight public transport isn't as good (the London tube stops running so you have much slower buses; most national trains stop running around midnight or shortly after). Google Maps directions now makes travelling via public transport much easier, due to being able to link different company/transport systems altogether.
2. I'd agree with other comments that there isn't much of a range in standard of living, but there is variation in cost of living, salary, available jobs.
3. The Royal Mail works well. Sometimes parcels/etc are sent by companies using private firms that are more of a pain (including due to their depots typically being miles away, so not feasible to pick up because you weren't in during their "anytime between 6am and 9pm" delivery slot - Royal Mail depots are much more common - also see point 6).
4. I use direct debit for all bills except my credit card, which I pay by debit card (direct debit is still possible for that though, this is my choice not to).
5. Cheques still exist but are increasingly rare (I only use them for B&Bs that either don't except a credit/debit card, or pass on the fee). Even for banking between individuals, we've had the ability for years to send money to someone between banks online, instantly (my understanding is this has been less common in the US). I also used to use cheques to pay my rent. One thing I will say about cheques is they're a right pain for me to receive (I have to go to the bank), but very easy to give (not as easy as card, but easier than me logging into my bank account, faffing with the security system, entering their account details and sending) - so if someone is willing to take a cheque, I don't mind writing them one.
6. Yes. Amazon in particular have collection places such as lockers in loads of locations, but I believe there are also more general pick-up points for other companies.
7. There's a reason the British are stereotyped for talking about the weather - it varies a lot. I noticed it rains a lot less now that I don't live on the coast (so bear in mind that the "it rains a lot" comments are likely dependent on where they live). A good thing is we don't have the extremes that some places do - sometimes it snows but we don't get loads; sometimes it's a hot summer but not massively so. The flipside is that therefore we're less prepared for it - a bit of snow causes problems; most homes don't have things like ceiling fans or air conditioning (though at least some offices will, especially if full of computers that might overheat).
9. In response to "However you do not need to be a UK taxpayer to receive treatment." - yes it's true that hospitals won't be refusing treatment or giving people bills, but note that non-EU immigrants now need to pay £200 per year for their visa duration (now 5 years before you can remain indefinitely) to cover possible health costs, on top of the usual visa fees, and on top of paying for the NHS through taxes. On a more general note, things in the UK have become much harder for non-EU immigration in recent years, and this seems to be a continuing trend.
12. I couldn't say what's typical. I get life insurance, private healthcare, long term sickness cover benefits at mine.
In Japan, the debit card, train pass, and credit card are all on the same card. I swipe my credit card to get onto trains and pay the bill later. How about there?
Credit and debit card are typically separate (how does it know which you want to pay with?)
Cards have been contactless for a few years. Although traditionally train passes have used tickets or cards, the London transport system now supports paying by contactless debit/credit card.
Pay everything each month and there are no extra charges, right?
This is true of my credit card at least (Barclaycard). Some accounts/cards may have monthly charges, but it's not the normal situation.