Originally Posted by Rocksteady

1) §61.51 Pilot logbooks.
(a) Training time and aeronautical experience. Each person must document and record the following time in a manner acceptable to the Administrator:

(1) Training and aeronautical experience used to meet the requirements for a certificate, rating, or flight review of this part.

(2) The aeronautical experience required for meeting the recent flight experience requirements of this part.

(b) Logbook entries. For the purposes of meeting the requirements of paragraph (a) of this section, each person must enter the following information for each flight or lesson logged:

(1) General—

(i) Date.

(ii) Total flight time or lesson time.

(iii) Location where the aircraft departed and arrived, or for lessons in a flight simulator or flight training device, the location where the lesson occurred.

(iv) Type and identification of aircraft, flight simulator, flight training device, or aviation training device, as appropriate.
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Key word there is "each person must document" So not Skywest, but you, and only if you need to count it towards ATP which you do because CTP is now required training towards ATP so it's required to be logged

2) no