In a precise dictionary, it appears as "時差惚".
But I have never sawn this. Most Japanese would use the Katakana.
(2) Why written in Katakana?
Simply because it is relatively new word to Japanese. Jet lag had been an unknown idea before aviation got popular.
Though "時差ボケ" is the composition of Japanese words "時差" and "ボケ（惚け）" , it might give impression to Japanese people that it is a foreign idea and thus should be written in Katakana. This conclusion is merely out of my conjecture, but it may be plausible to some extent.