I live in a small town in England. I also spent 10 years in Germany.

Do a large number of people live without a driver's license?
I have never had a driving license or a car. The good side is that I can live very cheaply, healthily and stress-free. The bad side is fewer girlfriends.

Can you shop and work without a car, even in a fairly small town/sparsely populated region? Can you travel easily to any major city without a car?
Yes. I can buy most things in my town. I can get to London, Canterbury and most other towns by train. Many cities in UK and Germany have extensive tram and bus networks that reach out to the suburbs

Are bike paths and foot bridges always a part of road planning (and rail planning, for that matter)?
For the most part - yes

Can people imagine living without a car, regardless of how much they might hate that fact?
When I tell people here that I don't own a car they tend to regard me as afflicted with some kind of illness - even when I tell them I can get to all the places I need to.

Well, my points were directed more to the difference between society and culture, but it is interesting to see that England, as often so, lives in both worlds - that is, European from history and current EU membership, cars from American dominated marketing.

I would have honestly thought that there would a certain number of non-drivers who are proud of that fact, as in Germany. But then, the level of awareness of vegetarian eating, as taken for granted in Britain (like the labelling, or the number of good places to eat where meatless menu items are a given) is pretty much unknown in Germany. It was always a problem finding somewhere good to eat with English speakers who were vegetarians, especially British women, since in Germany, about the only acceptable places to eat were Indian or Greek.