So are Jews today with the surname Kagan descended from the Khazars, and the rest from the ancient Israelites?
Probably not. I would suspect that the Jewish surname Kagan is related to the Kohanim-group names (Cohen, Kahn, Kohn, Kahane, etc.) Linguistically, it is possibly for a velar fricative (i.e. h) to become a velar stop (i.e. g), so this phonemic change probably what accounts for the name Kagan.
No. Kagan is the Russian pronunciation of the Hebrew word Cohen (priest). Russians always turn an "h" into a "g." It is also the Russian pronunciation of the Turkic word Khan. The Khazar word Kagan is the same as Khan, as in Ghengis Khan.
The Hebrew word Cohen, the Turkic word Khan and the Hawaiian word Kahuna probably all come from the same prehistoric root, along with the English word King.