Thank God I don't have to install Visual Studio 2003 onto Vista.
Well, that may be a bit premature. I have to make sure that debugging and the like will still work. But I don't see why it wouldn't. So yes, it does turn out that you can work on .NET 1.1 projects in Visual Studio 2005 with a couple of project templates called "Everett Compiler". The Everett Compiler project templates include Console Application and Class Library templates that use a custom MSBuild .targets file that directs the compiler to target the C# 1.1 compiler ("Everett") instead of the typical 2.0 compiler. The original theory was that if 2005 could be redirected to use the LINQ/C# 3.0 compiler, why couldn't it be redirected to use the 2.0 compiler?
When first creating and loading a C# 1.1 project, you'll see the following dialog:

As you will learn by clicking on "More Info," you can disable this dialog by adding a string value to HKLM\Software\Microsoft\VisualStudio\8.0\MSBuild\SafeImports with the value "C:\Program Files\Everett\Everett.targets" or "C:\Program Files (x86)\Everett\Everett.targets" if you're on x64 or Itanium systems:

Building ASP.NET support with this is somewhat more challenging. You should be comfortable with using the ASP.NET markup syntax, because you probably don't want to try to incorporate the designer changes in ASP.NET 2.0.
Having said so, there are some other things you need to do. The Everett projects by default build to a subfolder of \bin; you'll want to have them output to \bin by default. ASP.NET 1.1's code model is significantly different than 2.0: with 1.1: there is quite a bit less dynamic compilation done with codebehind projects in 1.1 than there is in 2.0.
For projects that I built up, instead of trying to open the original project (which you probably don't want to do), I copied and pasted the files into new projects in a different subtree of the directory structure, and set up that directory as the web site in IIS 7. Be sure to check out Justin Copeland's blog entry about setting up ASP.NET 1.1 with IIS 7 to avoid, you know, continually crashing IIS. :)