I need a custom script that would input information in a cell if input were entered in the row.

Yes, Use a custom calculation script in the "OperatorRow#" field. 

There are various ways to do this, but the general idea is something like this.

var oFldDate = this.getField("DateRow1"); 

var oFldFrom = this.getField("FromRow1"); 

var oFldTo = this.getField("ToRow1"); 

if((oFldDate.valueAsString != "") && (oFldFromvalueAsString != "") && (oFldTo.valueAsString != "") && ... etc ...)

    event.value = this.getField("User Name Field").value;

This will be a fairly long script because of the number of fields, and it will need to be on every "Operator" field. 

Two things you can do to simplify the script are

1) Create a more generalized document level function

2) Use a naming convention that will allow the script to acquire all fields in the row in a single go.  For example: Row1.Date, Row1.From, etc.

You would need a trusted function in a folder level script to access the Name parameter of the identity object (Edit>Preferences>Identity>Name) on the Acrobat menu.  You would then enter a custom validation script into every field in the row that sets the corresponding Operator field to the name of the user, conditional on whether the field has a value.  Example of trusted function:

fill_Operator = app.trustedFunction( function (oDoc,oFld)
{
app.beginPriv();
oDoc.getField(oFld).value=identity.name;
app.endPriv();
})

Example of the validation script:

if(event.value)
{
fill_Operator(this,"OperatorRow1");
}

If you use Right-click>Create Multiple Copies to create your fields you can end up with one validation script to copy to all fields with this tip.