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Preying on the Weak
This is a pretty simple mission. We need to kill the thugs that are shaking down the refugees. Just go out to the refuge camp and kill everything that's hostile. The thugs are just in groups of threes and they don't have any mages with them. You should have absolutely no trouble just crushing these guys. Take out the group in front, the group on the right and the one by the house near the exit.
Once they're dead, you will receive credit for winning the quest. Note that you don't seem to get a reward for this one, although the thugs do carry some decent equipment and money. Plus you're helping the poor.
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Out of Control
Talk to Ser Rylien inside the chantry to get more information about this one. She'll identify three apostates inside the city. You'll lose one point with Anders for taking this quest, at least if you treat the mages like they should be hunted. I wouldn't worry though, especially if you're planning to do his companion quest.
They are legitimately crazy though and really planning some dark things. You don't need to feel too bad. Just wander the city and look for them. They'll show up as rambling people. I found one on the left battlements, one by the entrance and another in the market. To start the fight, just follow them for a moment and keep talking to them. They should snap and attack. Each one is just a powerful mage. You should have no trouble surrounding them and interrupting their spells for an easy kill. At worst, they'll get off one strong spell and you'll have to use group heal. Nearby city guards should also help you which makes this all really easy.
You need to get their leader before the quest is over though. I don't know what clues the game thinks it gave you, but he's over in the back corner of the market. Just corner him and kill him. He's a little tougher than the others, but he's already trapped in a corner. Once he's dead, turn the quest in at the chantry for a 20 sovereign reward.
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Donation of Injury Kits / Donation of Poultices
This is really easy. Just have five injury kits in your inventory. Since Anders should have Cleansing Aura by now, you don't even need them. Hand them over for some easy experience.
They should then offer a quest to donate poultices. You probably won't find five potent poultices, so you'll need to make them. Have your herbalist craft them up with some ingredients you found or bought from innkeepers or merchants.
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From the Living Wood
This is a fairly simple hunting quest. You need to kill five Ancient Sylvan and get some of their wood. Just look around for them in the Wending Woods. None of them are before the bridge. Two of them are on the right side of the forked road. You have to walk through the darkspawn camp. They are in the back of the area, close to Ines the Botanist, if you have that quest. They should die pretty easily. They aren't much stronger than a normal one. Their nasty vine attack might be able to kill a mage or rogue, but your warriors will be fine.
The other three are in the other area. Two are close to the "middle" of the area. One is near the wounded milita soldier. He's just across from the lean-to in the camp. He'll probably be the last one that you kill.
Just turn in the wood at the chantry to get some experience and money. Note that Anders will not appreciate this that much. It will cost you an approval point if he's in the party. Not sure why though, since they are demonic trees.
There are several different quest boards to work off of for some easy money. Letters for the Commander, Merchants' Guild Board, Chanter's Board and Blight Orphans are all good ways to get some experience and money without much tricky combat.