Your observations are absolutely correct and it is not possible to focus an element placed in the headerTemplate. That template should contain only static HTML and it will not be possible to focus on the input element, without closing the popup. This limitation is due to the fact that when you blur the input element of the ComboBox, it will internally close the popup element (which is the expected behavior).
With the above in mind, please place the INPUT element from the headerTemplate outside the ComboBox widget.
Join us on our journey to create the world's most complete HTML 5 UI Framework - download Kendo UI