1. If you set up a default wifi system, then yes, you are likely to be open to people piggybacking of your connection. It really depends where you live though. People in cities are more likely to experience this than rural dwellers with no neighbours. However, many wifi routers now come with security enabled, and it really isn't difficult to set up basic security which will prevent people using your connection unless they are really determined. It's not that much more difficult to make you wifi network almost impossible for anyone unauthorised to be unable to access it.

2. No, if you find a free wifi hotspot, you are using their ISP to connect. Be cautious however, not all free wifi hotspots really want you to be using their connection.

3. It depends what address you want. .co.uk, .com etc
If you want a .co.uk for instance you will register via nominet http://www.nic.uk/ but you have to purchase the domain via an agent. These usually provide hosting for your website, and you really need to know what you are after before you start searching as costs and services vary dramatically. Just do a google search on domain registration for a huge variety of options. Once you've got a feeling for what you are after just come back and ask more questions - better in the lounge or even better in website feedback forum where the techie stuff goes on tho.