If you are considering a career as a web engineer, I think the first thing you need to ask yourself, do you enjoy tying and untying knots over and over again. If you do, you probably have the right stuff to be a really good programmer.
I've also heard web programming described as plumbing.
I wish I knew this before I started my Master's degree in IT. Programming is frustrating if you don't have an engineering background... you keep running into new problems and people telling you what you could do better.
I would never describe either finding new problems or people telling me what I could do better as frustrating! They are both some of my favourite things to experience!