I'll soon be managing a fleet of ASP.NET webservers...
What are the technical limitations with deploying an ASP.NET webserver? Can I just copy the file(s) over, restart the website, and I'm good to go?
Is this any different than what VS2010's "deploy" does?
Does it matter what "type" of project I have: a Web Site or a Web Application?
Web site or Web application doesn't matter. That is just a distinction that is made when the site is developed and compiled, not when it's deployed (except a web site will have more .dll files.)
You really can just copy all of the files over to your new website, as long as you are doing a clean copy. The application will automatically re-jit, so you don't necessarily have to restart the website.
To make things much easier, I would use an automated publisher, such as NaNT, Cruise control, or MSBuild. I've never used it, but the built in publisher from VS2010 may also give you what you need.
You may want to check out Web Deployment Projects (WDP) for Visual Studio. These are MSBuild-type projects that wrap functionality of aspnet_compile.exe into an easily configurable set of project properties. If you're doing automated builds and deployments, these types of projects make thing a lot simpler (at least they did for me!).
They provide you with the ability to swap out configuration files (appSettings, connectionStrings, etc.) depending on the type of build (Release, Debug, other) and you can also set how you want your output DLL's built (per-file, one assembly, etc.). This is very useful for web site applications -- not so much for web application projects.
If you have a mix of WSP's and WAP's, using WDP's may give you some consistency.
Check out Scott Guthrie's blog on using WDP's and also his VS2010 update.