In London, people do drive to work, but you can easily get by without a car due to the fantastic public transport system. If you do choose to drive, you should be aware of parking restrictions, and the very high toll to drive into the centre of the city during the week! Most people who work centrally use public transport (bus or tube) to work and those who live centrally would not require a car at all to get around. The tube and train network covers London very well. Biking is also becoming increasingly popular in London, especially with the recent introduction of 'Boris bikes', a cheap, user-friendly bike hire scheme.
EU members and those from commonwealth countries can exchange licenses on arrival in the UK. Within a year of arriving in the UK, American citizens will need to take a driving test through the DVLA (Driver and Vehicle Licensing Agency) to obtain a license and insurance - which are legal requirements for driving.
It is very easy to get around London by bus, overland train and the underground subway system (the tube) which covers most of the city and surrounding areas.
Yes, there are frequent coaches/buses and trains to other locations in the UK and to the areas surrounding London.
No, although it is wise to be aware that pickpocketers operate in public transport areas, and you should try to avoid displaying your wallet or phone to others. London is generally a very safe city to get around and at any time there will usually be hundreds of people making their way around alongside you. The streets are mostly brightly lit and well maintained, and vehicles are usually respectful of cyclists and pedestrians - especially in the city centre.
For both the overland and underground train, you can obtain tickets at the station. You may also buy train and bus tickets at local newsstands, online or by phone. It is recommended you buy an Oyster card when you first arrive as this will greatly reduce the cost of your journey. Most people buy a weekly, monthly or yearly ticket to save on costs.