as the combobox never goes to H3 the code for combobox keypress is never called you would need to add code in selectionchange event
try this change
vb Code:
  1. On Error GoTo errHandler
  2.  
  3. 'If Target.Count > 1 Then GoTo exitHandler-- why did this kill it for me?
  4. If Target.Address = "$H$4" Then Range("d12").Activate: Exit Sub
  5.   Set cboTemp = ws.OLEObjects("TempCombo")
  6.     On Error Resume Next

the double tab effect is because cells in cols e and f are not locked so can be selected under the combobox