Merrill/Approval

Friendship
Merrill's friendship can be gained by:
 * Siding with the mages or the elves in quests.
 * Choosing options that favor blood magic or demons.
 * By being agreeable to her in conversation.

Act 1 Enemies Among Us: When rescuing Keran, if carver is present, choosing "Templars are just as bad."
 * Act of Mercy: Tell Grace that you will distract the Templar (Thrask). Choose the agreeable options and then "Murder is too far." This will earn you (Confirmed)
 * An Errant Letter: Keep Ser Thrask's secret
 * Bait and Switch: Defend Merrill (Call on Companion) at the end of the quest
 * Enemies Among Us: For, choose the following dialogue options while speaking with Knight-Captain Cullen in The Gallows:
 * "Not all mages are like that."
 * "You're wrong."
 * Finders Keepers: Attempt to trick the guards by the docks with the special choice, "Fire!" for (Available only for a Humorous/Charming Hawke)
 * Note: It is also possible to gain the same approval for Varric, but you cannot receive approval from both characters. If Varric and Merrill are both with you at the time, Merrill will always receive the approval over Varric.
 * The Gallows: Visiting the Gallows with Fenris for the first time will initiate a dialogue scene. If Merrill is present, choose the response, "It is wrong to oppress mages." for
 * Magistrate's Orders: Kill Keldar
 * Shepherding Wolves: Accept the quest from Sister Petrice Shepherding Wolves: Refuse to hand Ketojan over to the Qunari
 * Tranquility: Promise Lirene you won't harm Anders
 * Wayward Son: Tell Vincento to make it up to his son for his neglect then go to the Gallows and tell Thrask you want to help find Feynriel to gain an additional  Wayward Son: Allow Feynriel to go to the Dalish
 * Welcome Home: For, choose either of the following dialogue options:
 * "You're welcome."
 * "Did you really?"

Act 2
 * A Bitter Pill: For, when asked by Orana what she should do, select the following dialogue options:
 * "Come work for me."
 * "I will pay her."
 * Forbidden Knowledge: Choose to use or take each evil tome in the sidequest: for each tome not destroyed while Merrill is in the party
 * Mirror Image: Flirt with Merrill Mirror Image: Accept the quest  Mirror Image: Give her the Arulin'Holm, either on the first or the second chance
 * Night Terrors: Tell Marethari that you will not kill Feynriel:
 * Night Terrors: Accept the sloth demon's offer to bring him Feynriel:
 * Wooden Halla: If you are already on the path of friendship with Merrill, then any response will earn you a

Act 3
 * A New Path: When talking to Merrill to initiate the quest, agree without suggesting she talk to the keeper first: A New Path: Ask Marethari how to fix the eluvian:  A New Path: After the battle with the demon tell Merrill that Marethari was stupid:

Rivalry
Rivalry with Merrill can generally be earned by contradicting the "hows" of developing a friendship with her, as mentioned above.

Act 1
 * Act of Mercy Telling Grace to go back to the circle:
 * Bait and Switch: if at the end of the quest Hawke says, "I'm well aware." (only available to a non-mage Hawke who brought a mage along during the second half of this quest.)

Act 2
 * Wayward Son Convince Feynriel to go the Circle:
 * The Gallows Visiting the Gallows with Fenris and Merrill for the first time will initiate a dialogue scene. If Fenris is present, choose the response, "The Circle is necessary."
 * Welcome Home "Don't Thank Me":
 * Magistrate's Orders Choosing to help Keldar:
 * Enemies Among Us When rescuing Keran, if Carver is present, choosing magic is not the problem.
 * Forbidden Knowledge Destroy each evil tome in the sidequest: for each tome destroyed while Merrill is in the party
 * Night Terrors During the dialogue with Marethari, choose to kill Feynriel if things go wrong in the fade:
 * Night Terrors Refuse the sloth demon's offer:
 * Night Terrors During the dialogue with the Marethari-demon choose the "Don't trust it Merrill" option:
 * Night Terrors During the dialogue with the Marethari-demon choose the "Don't dare turn on me" option:
 * Mirror Image Refuse to do the quest. Refuse via witty + investigate: . Refuse aggressively: . The quest will still be available, though.
 * Mirror Image: Tell Merrill that Pol ran away from her because he is a coward (aggressive option), after defeating the Varterral:
 * Wooden Halla If you are already on the path of Rivalry with Merrill, then any response will earn you a
 * Back from Sundermount: After NOT giving the Arulin'Holm to Merrill (as the Keeper requests), tell Merrill that you are trying to help her (agreeable tactful option), or that she can't complete the mirror (wry, humourous or witty option): You get for either of these. Telling Merrill you don't trust her with the tool (aggressive option) instead, only nets you.

Act 3
 * A New Path: When talking to Merrill to initiate the quest, suggest she talk to the keeper first:
 * A New Path: Telling the rest of her clan that you will take responsibility for Merrill's actions:
 * A New Path: After the battle with the demon, tell Merrill that she should have listened: