Has it been reported from the Prophet (peace be upon him) anything specific regarding the sitting of rest (jalsat al-istiraha) for the khateeb, especially what some khateebs do by saying "Supplicate to Allah while you are certain of a response" before sitting for it? And are hands raised for amen during the khateeb's supplication at the end of the sermon?

There is no specific supplication authentically reported to be recited between the two sermons of Jumu'ah, neither for the Imam nor for the congregants. However, it is permissible to supplicate for the good of this world and the Hereafter during this period, hoping to coincide with the hour of response on Friday. It is also permissible for the congregants to say "Ameen" to the supplication of the Khateeb (preacher) without raising their hands.