Working in Web Design is a great choice. There are many ways to contribute and many ways that the work get done. So easier questions first:
- Web design work is available every where so no relocation is necessary but on the other hand if you want to relocate you can.
- Normally there is not much travelling until you get into a senior position and need to visit clients.
You can work by yourself as a freelancer, in a web design company or as part of a big team in a multinational company. I suggest you start your working career in a smallish web design company where you will get the chance to try your hand on a wide range of skills. In the UK most people start in web design after going to university.