Is there an IDE that highlights each HTML closing tag (as you highlight over the opening tag, for example), similar to how many IDE's highlight where the opening/closing braces are for functions and such?
Yes lots, you can use notepad++, aptana if you have a better machine, pspad
if you can use aptana and add the ftp site so you can have full access of everything ;)
There are quite many IDEs that can do that and it's only up to you which one will you choose. For example I often use NetBeans and Eclipse for HTML+CSS and PHP, sometime VisualStudio (for ASP.NET). For very fast editing, or something small I use Notepad++. All of them supports highlighting closing and beginning tags when marked. Also Dreamweaver is good for HTML, but I prefer to use OpenSource or freeware software (except for Visual Studio, but Express editions are also available free of charge).
So as you can see there are many options. I have to say that Notepad++ is quite powerful tool, very hight customizable, but for bigger HTML projects I use NetBeans due to it's code completion, syntax highlighting and other useful tools.
Regards, Ventus