Is it possible to add calendar and Time inside an edit event popup

DayPilot includes a DatePicker control which you can link to a Label or TextBox control.

<DayPilot:DayPilotDatePicker 
        runat="server" 
        ID="picker" 
        ClientObjectName="picker" 
        TargetID="LabelDate" 
        ImageUrl="../Media/linked/datepicker.png"
        Text="Change"
    ></DayPilot:DayPilotDatePicker>

This is an example from this demo page:
http://www.daypilot.org/demo/Calendar/DatePicker.aspx

The source code of the Edit.aspx is in the DayPilot package (see Demo/Calendar/Edit.aspx and Edit.aspx.cs) and you can modify it as needed. There is no problem with splitting the datetime field into two - they are simple TextBox controls. The date is assigned in Page_Load:

protected void Page_Load(object sender, EventArgs e)
{
    if (!IsPostBack)
    {
        // ...

        TextBoxStart.Text = Convert.ToDateTime(ev["start"]).ToString();
        TextBoxEnd.Text = Convert.ToDateTime(ev["end"]).ToString();
    }
}

You can download the trial package here:
http://www.daypilot.org/try/