Many do. There are several reasons they're not very competitive. They were late to that market, just following Uber's coattails. Their advertising is not as slick and the apps often have poor UI, so it's not as enjoyable and effortless to use. And significantly, they actually follow the law, which results in higher costs to the end-user than law breakers like Uber.

In some UK cities and in New York there is one. It's called Gett (http://gett.com/uk/). It's more expensive and the experience is worse.

Probably not. What makes Uber/Lyft so compelling is that they can do two things cab companies are legally not allowed to do: (1) subsidize fares with VC money; and (2) cross-subsidize between surge fares and fares during periods of low demand.

From my office in the Watergate, it takes a good 5-7 minutes to call an Uber/Lyft (some GPS/app bug causes people to drive right by and end up on Rock Creek Parkway). If it wasn't $3 to my apartment in Gallery Place, I'd just grab a cab from the cab line.

Several markets have this (several also have taxis IN uber app). They often fall back to cash payments.

Curb is the app I've seen for Taxi service.