Is there any reason why I shouldn't install Java? I know that you can improve performance and battery life by not having Flash on your Mac. Is there any similar advantage to not installing Java?
Installing java won't have much effect on battery life. 

I don't think installing Flash will have much effect either.

Where the issue is is when you hit a website with badly written flash (and to a very much less extent Java) where the animation takes up CPU but you can just leave these sites - you will be no worse off than not installing.

Java is now rare on websites and not really used for parts of the web site the user does not need.

I think a better way is to install a plugin for your browser like ClickToFlash which stops Flash loading until you tell it.