Is it worth going to university to study website design?

Of course. Two reasons. Opens your mind to many other opportunities and provides credibility with employers. Also, a person can and should freelance during school for both money and experience

if you can hookup with a good company to design for the yeah go for it

no technoalegy changes all the time by the time your in your last year the software will have changed and the software you used in your first year will be out of date

i think you should get a job in it that way you will have experince and will be using upto date software

or start you own business why now more money for you

If it’s what you are passionate about, YES.