Do I need a degree to become a web developer?

The short answer is no. These days, having a degree in computer science is not a requirement to find a job in web development. There are many ways to teach yourself online or offline to acquire the knowledge needed for starting out as a developer.

o.....................................

Absolutely not.  Better would be to display samples of your work on your own website.

Nope.  I started developing websites for people years ago and my degree is not in web development...it is in marketing.

That depends on whether you want to start your own business or work for somebody else.  You might have difficulty getting a job for somebody else since you would be competing with people who have the education.