On the note of formal education, I would say it has advantages. I am trained for network management. Since starting my business, I may never use officially use that education, and I have never yet formally worked as a network admin, but it has given me a serious advantage as a web designer.
With what I now know, I think a decent understanding of networking should be required for web design. After all, the web is really just one huge network that operates on the exact same principles as a Local Area Network (LAN). It would do web designers well to understand it.
I also agree that learning Flash is helpful, but not necessary. If you do learn Flash, you need to be careful not to overuse it. Flash requires a lot of time to learn well, and most Flash development is very time consuming. When you invest all that time and energy, you might be tempted to use it just because you can, or because you feel like you need to because you spent so much time learning it. Don't let Flash hinder your ability to know what's best for the client.
I also agree that Photoshop rocks. It is the only Adobe program I actually like, but I would also recommend a copy of Fireworks before Adobe puts their own spin on it with the next version. Work with both so you get familiar with the differences. Each one has some things it does better.