Friendship with Sebastian can generally be earned by encouraging him to stay with the Chantry, by siding with the Chantry and the templars and by being helpful to those in need.
Dissent: Hawke stops Anders from harming Ella (+15). An additional (+5) if Hawke then says, "Just what it looked like." upon meeting Ella outside the tunnels, and (+5) if Hawke then tells her to return to the Circle.
Night Terrors: Tell Marethari "I understand" (that Feynriel must not become an abomination). (+10)
Note: To avoid getting this point negated by another (+10), start the quest without Sebastian. Postpone the quest when Arianni asks if you are ready. Then, go to Merrill's house, put Sebastian in your party and resume the quest.
Offered and Lost: Warn Elthina ("Someone used your seal") (+10). Note that the option "You're useless" in the same conversation will yield rivalry points, if it wasn't chosen earlier in this Act.
Offered and Lost: Tell the Viscount not to hide what happened (+10)
Prime Suspect: If Hawke begins the conversation with Emeric along an aggressive or humorous line of conversation until Sebastian comments that "this is a worthy cause." and then finishes with the diplomatic option to accept the quest. (+5)
Rivalry with Sebastian can generally be earned by pushing him to retake Starkhaven, by siding against the Chantry and by making aggressive, disagreeable or otherwise "dishonorable" choices.
Fool's Gold: Tell Emrys that you'd rather have the weapon than save his brother (+10)
PC 1.04 - no approval change with this choice
Grand Cleric Elthina: Choosing "You're useless" when speaking to her will net (+5). Note that you will gain his friendship if during Offered and Lost you will also choose "Someone used your seal."
Note: Approval will only count if a small cutscene of Sebastian disapproving of the quest is shown. Sometimes it is not shown, depending on which characters you have in the party.
Prime Suspect: Picking all the aggressive responses when talking to Emeric (+5)
Questioning Beliefs: When speaking to Sebastian, the first and second set of choices all take you the same place. In the third set, choose Starkhaven (+15) (If you maxed Friendship, he will not gain Rivalry)
Repentance: When speaking to Allure, choose "Don't listen to her" (+15)
Repentance: When speaking to Flora, choose "Don't try to wiggle out of this" (+5)
Repentance: During the final conversation back in the Chantry, choose "You must take back your land." (+10)
Faith: In the final conversation when Grand Cleric Elthina refuses to leave, reply "Fine, your choice" (+15)
Haunted: Let Varric keep the shard of the idol (+10)
A New Path: In the beginning of the quest, if you force Merrill to talk to Marethari without asking by just talking to Marethari beforehand, and you pick the option to ask Marethari to help fix the eluvian (+5)