Quite a few years ago I made the decision to graduate myself from online forums and stick to lurking from afar (just no longer had the time or energy to get involved in the forum micro-environment any more). Thus, I've been lurking these forums, stalking and reading discussions for a looong time. And I've not logged in or made a single post, until today. 
This topic piqued my interest enough to break my covenant just this once. So kudos to OP.
Also, if it makes any difference, I'm coming mainly from the perspective of a Dalish Warden, Mage Hawke, and Dalish Inquisitor.
All Female.
1. Things I dislike about characters I adore with all my heart:
This is actually really hard. Most of my dislikes for loved characters are just me complaining about lack of certain in-game functionality or unrealistic things I wish I could do (i.e.: fantasies unfulfilled *sigh*). Also there's so many characters I love but I'll just put the ones that I can actually list something I dislike for.
DAO
Dog - That he takes up a character slot. I desperately want to take him everywhere but sometimes he has to stay at camp.
Wynne - That she didn't knit my Warden a griffon onesie.
DA2
Aveline - The fact that she'll never love my Hawke the way my Hawke loves, covets and worships her. *cries in corner with Aveline shrine*
The Sabrae Clan - I love em but dammit where's your solidarity you fools! My Warden would not be proud of the way y'all did in Merrill.
DAI
Minaeve - I dislike that she gets replaced by Tranquilady. I really liked pestering talking to her and really wanted to be able to flirt with her.
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2. Things I like about characters I loathe with most of the fibers of my being:
DAO
Morrigan - I like that even though I dislike her in so many ways, she is super appreciative of the Warden's friendship. If you work with her prickly nature instead of against it, she expresses how being a friend means a lot to her. I tend to take a "Cool story bro." and "Relax Morrigan" attitude with her and somehow she likes me. I dunno, I like that she values me being able to put up with her. lol
Broodmother? -- Like ew. But I admire its...uhh....determination?
sorry I'm gonna puke even talking about it.
Zevran - I wouldn't say I dislike him 100%...but I just don't like him much at all. He's my least fav companion. But hmm...I like that he's not super tall? I guess.
Cullen - His hair?
DA2
Fenris - I so desperately want my Mage Hawke to buy the heaviest staff she can find, name it "intersectionality" and gently beat him over the head with it until he gets it. That dream aside, I do like the fact that you can at one point at least call him out on his hypocrisies and he just kinda chuckles and says 'oh yeah'. I never romance him so I dunno how much more you can do it.
Anders - Couldn't find anything about his personality or views that I like (I can vaguely empathize...but he has zero sympathy from me especially for his final actions). But regardless, the one thing that I will always absolutely LOVE him for is the fact that he can save my adorable baby brother Carver (
) from death. Anders gets a huge +all-my-approval for that single thing.
Isabela - I'm not a huge fan of her personality or humor. I don't mind her views...but I just don't like how her actions often lead to bad outcomes -- I also don't like how she got mildy annoyed that I turned her down. Like plz. Actually, I was on the verge of being peeved enough to actually kick her out but she did ONE thing that completely redeemed her for me. Let me just say that Carver is my absolute fav DA2 character. I adore him in every way -- he can hate my guts, spit on my shoes and I'll still love him forever and ever. Now on my first DA2 playthrough, the one time I decided not to metagame was for the deep roads expedition. Needless to say I didn't know I had to bring Anders along, and my baby brother died. It made me distraught enough to nearly rage quit the game -- The game had already pissed me off enough and killing Carver was the last straw. My Hawke had been pushed to the edge and was literally ready to give up dealing with everyone else's bs and wreak havoc on all of Kirkwall herself (I didn't care if I had to make an Everything-and-Everyone-in-Kirkwall-is-Vaporized mod to do it). Enter Isabela. I already hated her guts and she was to be my first victim I swore...but then she was the first one to turn to Hawke and say the 'i heard what happened in the deep roads with (insert sibling) and I'm sorry.' line. Even when everyone else immediately started going on about their own problems, she was the only one to take the time to say that. Isabela! of all the darn people, not even my LI. But in that one line, she completely talked me and Hawke off the metaphorical edge of no return. Kirkwall literally had Isabela to thank for Hawke's continued assistance. Perhaps Isabela could have talked Anders off the edge too. >__>
Cullen - His hair?
DAI
Cullen - His armor. Dat lion helmet tho.
Sera - Highly bombastic and opinionated...an extremely off-putting mix that I have a hard time getting down with. Romanced her with my Dalish Inquisitor instead of a Qunari specifically to make it hard for her even at the cost of it being equally as taxing on myself to find love with her. But I LOVE LOVE LOVE that one moment where you can ask her to marry you and she metaphorically, completely sobers up for a split second and calls you crazy. Just the way she looks at you and her serious tone in that one moment. I dare say that is probably THE SEXIEST moment in the entire game for me -- definitely my fav Sera moment. Makes me completely weak in the knees! *sigh* if only I could see more of THAT Sera...
Actually, that's probably what I similarly love about Merrill the most. Yes she's the most adorable, word bumbling socially-inept thing I've ever seen...but hot damn when she turns into a beast in battle (yelling all those things in Elvhen in that slightly deeper voice) my Hawke just gets all warm in...places.
Must be some kind of thing I have for people who act one way on the outside, suddenly displaying the complete opposite.
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Note:
Characters I love that are immune to my ire: Carver, Leliana, Merrill, Vivienne.
Characters I hate that will never get my love: Celene.....Celene.....Celene.