Partial answer to: reference for Katha as pupil of Vaishampayana and expounder of Shaka of Yajurveda:
Katha was a pupil of Vaishampayana who got Yajurveda from VedaVyasa as presented in this answer.
According to Patanjali's Mahabhashya 4.3.104:
यदयं कलापिखाडायनग्रहणं करोति तज्ज्ञापयत्याचार्यो नान्तेवास्यन्तेवासिभ्यो भवतीति।
कथं कृत्वा ज्ञापकम्? वैशम्पायनान्तेवासी कठः, कठान्तेवासी खाडायनः। वैशम्पायनान्तेवासी कलापी।
यदि चान्तेवास्यन्तेवासिभ्योऽपि स्यात्कलापिखाडायनग्रहणमनर्थकं स्यात्। पश्यति त्वाचार्यो नान्तेवास्यन्तेवासिभ्यो भवतीति, ततः कलापिखाडायनग्रहणं करोति।।
vaiśampāyanāntevāsī kaṭhaḥ means "Vaishanmpayana's Shishya Katha"
Quoting English translation from Volume 4:
The pupils must be direct ones, and not pupil of pupils. For had that been the meaning, Kalapi being a pupil of Vaisampayana, the words 'pupils of Vaisampayana' would have included the 'pupils of Kalapi.' But that is not so, the word Kalåpl being separately mentioned in the satra. So also Katha is a pupil of Vaisampayapa
, and Khadayana is a pupil of Katha and had the sätra applied to pupil's pupil also, then the affix for would have applied to mist97 also. But that is not the case, Khadayana being mentioned.. in Saunakadi class. (IV. 3. 106).
Katha is a Shakha of Krishna Yajurveda enounced by Katha and Katha Upanishad is associated with that Shakha.
"Katha" is mentioned in the source scripture Ashtadhyayi of Panini also:
४.३.१०७ कठचरकात् लुक् ।
4.3.107 kaṭhacarakāt luk ।
Which is explained by Mahabhashya that Kathah refers to a person who study Chhandas (meters) enounced by Katha. Hence it is clear that the word Katha in "Katha Upanishad" can refer to the name of sage "Katha" by whom Katha Shaka is enounced.
Note: Etymology of Katha, as told by sanskrit dictionary being "distressed" may be applicable for the naming of sage Katha and for knowing that, the study of story or biography of sage Katha is required.