DayPilot is compiled with .NET 2.0 and you can use it on IIS 6 with .NET Framework 2.0. Everything should work.
Just note that most of the tutorials at http://code.daypilot.org are built using Visual Studio 2010 or higher and use .NET Framework 4.0. The demo web application in the DayPilot package is using .NET Framework 2.0 so you can start from there.
1. You can copy the web.config from the DayPilot demo (Demo directory of the zip package). This version of the web.config is compatible with .Net Framework 2.0.
2. The actual .NET Framework version that will be used for the web application is determined by the web server. Make sure that the web application in the IIS is set to use .NET Framework 2.0.
Right-click the application in IIS manager -> Properties -> ASP.NET tab. Choose the ASP.NET version. Make sure that all websites in the same app pool use the same .NET version.
During development you should set .NET Framework 2.0 as the "Target Framework" in Visual Studio: Right click the web app -> Property Pages -> Build tab.