1. I don't think it would be easy to get around without a car. They're putting in a ton of light rail tracks but I think it's going to be several years before anything is finished -- right now there is a light rail, but it's very limited and I haven't found it useful at all. You can take the bus but I think it takes a long time to get anywhere that way.
2. Coming from Chicago, I think housing in Denver is incredibly reasonable. You can buy a house in the city for under $200,000 and rental prices are not bad at all -- we paid $850 a month for a cute little two bedroom in a hip neighborhood.
3. If you're not familiar with what neighborhoods are where,
here is a good map of neighborhoods. In terms of what is a good neighborhood, it really depends on what you consider "good" -- some people really like Stapleton, for example, but it feels way too suburban to me. My favorite neighborhoods are Baker (beautiful, old Victorian houses and apartments, decent bars within walking distance), Five Points (diverse and cool), Capitol Hill (somewhat trendy with mostly larger, old apartment/condo buildings, lots of bars), Highlands/West Highlands (kind of hip but small, cute little shops and restaurants, compare to Bucktown in Chicago but less upscale), Wash Park (more established, upscale neighborhood with gorgeous large park where everyone goes jogging), LoDo (right by downtown, lots of lofts [expensive], restaurants and breweries), and Congress Park (pretty houses). Cherry Creek is where the upscale mall, shops and restaurants are and Country Club has huge, gorgeous houses (it reminds me of River Forest if you're at all familiar with Chicago). I haven't been here for very long so there are many neighborhoods I don't know anything about, but I know most of the central ones.
Let me know if you have any more questions. I love talking about Denver!