What's your favorite mission/area across all the DA titles?

DAO
Favourite area would have been the deep roads. Specifically the build up to the broodmothers lair. Very creepy and they built up the suspense really well - it gave more context to how horrifying and cruel the blight is. The rest of the deep roads were a bit of a slog however.
Favourite quest would be the battle of denerim i think. It was the perfect climax, everything you had done had led to this moment. All your companions fought by your side and you saw how ruined each of denerim's areas had become.

DA2
Favourite area is pretty tough for obvious reasons. I dunno honestly.
Favourite quest would be All that Remains. I cry every time.

DAI
Favourite area would be Crestwood. It's the best paced area in the whole game, and i liked the mayors backstory.
Favourite quest is in your heart shall burn. It's the inquisitions first (and only) loss. You find out the extent of what you are fighting in the worst way possible.

DAO: Favorite area: Denerim.
         It`s hard to choose only one favorite quest so i`ll choose the ones attached to Redcliffe (attack at Redcliffe+the Arl of Redcliffe+The Urn of Sacred Ashes) and all the Origin quests.

DAII: Favorite areas: Lowtown and Hightown.
         Favorite quest: Legacy.

DAI: Favorite area: Crestwood.
        Favorite quests: In Hushed Whispers and Here Lies the Abyss.

I've never bought any of the Dragon Age DLC's. What was the quality of them? If you need me to be more specific, how about Inquisitions DLC?

DAO: Favorite area is probably Orzammar in general. The contrast between the Diamond Quarter and Dust town, the pub with that drunk dwarf who's muttering either dwarvish or some drunkard language, and especially the town criers throughout the quest arc all set the atmosphere for me. My favorite quest is probably Paragon of her Kind, specifically Hespith's chanting up to the broodmother reveal. Very creepy.

DA2: Hmm favorite area is tougher here. Probably Lowtown, but more because I associate it with Hawke's time in the slums. Favorite quest is the Act 2 finale starting with the confrontation with the Arishok in his compound.

DAI: Haven't played.

Dragon Age Origins
Quest: The Landsmeet
Area: Circle Tower

Dragon Age II
Quest: Demands of the Qun
Area: Hawke Estate

Dragon Age Inquisition
Quest: Wicked Eyes and Wicked Hearts
Area: Skyhold

Skyhold would have been better without all the loading screens, and having to open 900 doors to get anywhere.

DAO Quest: first day the come and catch every one..        A Paragon of Her Kind. I loved so many quests in DAO but this one had the right balance of atmosphere, interesting fights, lore and drama.
DAO Area: This is tougher but ultimately I'd say Orzammer was a really great area, espeacially for a dwarven noble - every conversation upon coming back gets you in the heart place.

DA2 Quest: All that Remains, first time a video game made me cry
DA2 Area: Chateau Haine & surrounds

DAI Quest: What Pride Has Wrought, with Wicked Eyes and Wicked Hearts rounding a close second.
DAI Area: The deeproads/titan areas in Descent were gorgeous! And I loved investigating and uncovering lore in them. This dlc convinced me it was possible to have a dwarf/deeproads based main Dragon Age game.

Origins: Denerim... or Orzammar. Maybe the Brecellian Forest. Or wait, uh, Redcliffe. It's hard to pick a single favorite. The tone and the look and feel of all the locations are so superb and have so much of that Dragon Age-ness. I will tentatively say that Denerim is my favorite. And my favorite quests are either Broken Circle or A Village Under Siege in Redcliffe. Both these quests had what I love most in fantasy: huge scale, desperate circumstances and an epic confrontation with the supernatural. The tone, pacing and action in both these quests is superb. Especially Broken Circle, I don't think Dragon Age has a quest that feels as epic or that left me feeling as fatigued and accomplished.

Inquisition: The Frostback Basin is probably my favorite area. It actually felt relatively alive, which I wouldn't say for most of this game's locations. Loved the Avvar and how they filled the space. The environment itself is lush and full of things to take a stab at. Even though I kinda hate fighting spiders every ten feet.

My favorite quest is harder. There are a few really epic ones. In Your Heart Shall Burn was pretty damn great. The epicness of the looming confrontation, the strange appearance of Dorian/Cole. The overwhelming darkness that engulfs the main narrative as we really get a taste of what we're up against. And the major shakeups that follow. In Hushed Whispers was like something out of a comic book. Like one of those ultra-dark alternate universe stories they do every once in awhile where Joker is a god-emperor or Superman is destroying the world. To see the actual stakes of the war with Corypheus was a great rally and really injected some more urgency into the main plot. Finally, What Pride Had Wrought is where I actually felt like the for real actual goddamn Inquisitor for the first time. Hunting my enemies, leading an army into battle. Magicking up and down the forest stream, leaving destruction in my wake - and then the temple of Mythal as everything that good fantasy promises. Ancient Alien Secrets and things you thought were long dead.

Hmm. Yeah, I'm gonna go with WPHW as my favorite quest

Ohhh no, it's probably The Descent honestly. Yeah it's the Descent.

DAO
quest - fade 
area- camp

DA2
quest - loose ends
area - The Hanged Man

DAI
quest - Tomb of Fairel
area - Hissing Wastes

There were at least a couple of favorite missions and areas in each games for me.

In DAO,
For mission, Redcliffe siege. I was so pleased to see all the people I have recruited or forced to join the fight were present in the fight. It made the experience a lot bearable amidst all those creepy undead coming from every directions. I was also relieved that they all survived. Due to Morrigan/Wynne spamming heals, of course.
For area? The Deep Roads. It was an amazing and scary experience. I felt the overwhelming sense of venturing into a horrifying territory.

In DA2,
I guess it has to be Mark of the Assassin areas. I loved the parties and infiltrating the chateau at night.
For mission? Forking All That Remains! That mission made the game totally unplayable for me. I was totally depressed for a few days.

In DAI,
Clearly it is difficult to choose between Wicked Eyes and In Your Heart Shall Burn. I loved the atmosphere of the Winter Palace at night during Wicked Eyes. The music was also ethereal. I will never get bored of that mission. In Your Heart Shall Burn beats it out in terms of everything, I guess. After closing the breach, they started to celebrate. It was all feel good experience. And then, the hell broke loose with enemies started appearing with the torches. I had the immediate flashbacks to Redcliffe siege! The night setting made it more dramatic. It was a montage of intense experience after intense experience. I lost it when Mother Giselle started singing The Dawn Will Come, then Skyhold journey came and I went mad. And that music has be the best a DA game can offer.
The Deep Roads in The Descent dlc might be my favorite area. Even though it wasn't like the Orzammar's Deep Roads areas that gave me the overwhelming sense of dread, it was replaced with the wonder of exploring into a totally unknown depths. The Vestibule of the Pure had me get my jaw dropped.