OK. It seems to be working for me.My steps:
1. I've extracted DayPilotProTrial-3.5.2.zip/Binary/DayPilot.dll into c:\components\daypilot-3.5-trial
2. I've opened VWD and created a new web site in the filesystem.
3. I've added DayPilot (c:\components\daypilot-3.5-trial\daypilot.dll) to the toolbox according to http://www.daypilot.org/installation.html.
4. I dragged DayPilotCalendar from the toolbox to Default.aspx in the design mode.
5. I double-clicked TimeRangeSelected event in the properties window. The following code was created:
using System; 
using System.Data; 
using System.Configuration; 
using System.Web; 
using System.Web.Security; 
using System.Web.UI; 
using System.Web.UI.WebControls; 
using System.Web.UI.WebControls.WebParts; 
using System.Web.UI.HtmlControls; 
public partial class _Default : System.Web.UI.Page 
{ 
protected void Page_Load(object sender, EventArgs e) 
{ 
} 
protected void DayPilotCalendar1_TimeRangeSelected(object sender, DayPilot.Web.Ui.Events.TimeRangeSelectedEventArgs e) 
{ 
} 
} 
6. I've changed TimeRangeSelectedHandling to PostBack.
7. I've added a new Label control to the page and changed the DayPilotCalendar1_TimeRangeSelected method as follows:
protected void DayPilotCalendar1_TimeRangeSelected(object sender, DayPilot.Web.Ui.Events.TimeRangeSelectedEventArgs e) 
{ 
Label1.Text = e.Start.ToString(); 
}
8. Pressed F5 and it works.