Great post on alternatives and more reliable methods than ".Select". After reading, you can adjust your code. How to avoid using Select in Excel VBA
To answer your question, replace
Selection.EntireRow.Cut
with
Range(Selection.EntireRow, Selection.Offset(1, 0).EntireRow).Cut
This should get you a good start, you'll need to add some code to not cut all 5 rows above if some of the are blank because they've already been cut or you could remove blank rows on sheet 2 once this code is done.
Sub GetDipsData()
Dim i As Long
Dim c As Long
Dim LastConsecutiveDip As Long
Dim vLastRow As Long
Sheets("Sheet1").Activate
vLastRow = Cells(Rows.Count, "E").End(xlUp).Row
Application.ScreenUpdating = False
For i = 2 To vLastRow
If Cells(i, "E") = 1 Then
s2LastRow = Sheets("Sheet2").Cells(Rows.Count, "E").End(xlUp).Row
For c = i + 1 To vLastRow
If Cells(c, "E") = 1 Then
LastConsecutiveDip = c
Else
Exit For
End If
Next
If c <> i + 2 Then
'copy 5 above and 5 below
If i < 6 Then
Range(Rows(2), Rows(c).Offset(5, 0).EntireRow).Cut Sheets("Sheet2").Range("A" & s2LastRow)
ElseIf c + 5 > vLastRow Then
Range(Rows(i).Offset(-5, 0), Rows(vLastRow).EntireRow).Cut Sheets("Sheet2").Range("A" & s2LastRow)
Else
Range(Rows(i).Offset(-5, 0), Rows(c).Offset(5, 0).EntireRow).Cut Sheets("Sheet2").Range("A" & s2LastRow)
End If
i = c + 5
Else
'just copy 2 rows
If i + 1 > vLastRow Then
Rows(i).Cut Sheets("Sheet2").Range("A" & s2LastRow)
Else
Range(Rows(i), Rows(i).Offset(1, 0).EntireRow).Cut Sheets("Sheet2").Range("A" & s2LastRow)
i = i + 2
End If
End If
End If
Next
Application.ScreenUpdating = True
End Sub