Favorite: Variable Player Powers. Co-op or competitve, it doesn't matter. Anything that means you have to gear a strategy towards what you are playing rather than a generic strategy for everybody. [TI3, SotM, Netrunner, Ogre, Myth]
Least favorite: Player Elimination, no question.
But that's low-hanging fruit, so:
Second-least favorite: Commodity Speculation [Merchant of Venice, Navegador, Glen More, Archipelago]
Favourite: Co-Op Games. Followed by Player Characters with Special Skills (rather than just being 'Green' or 'Yellow') and by Narrated Encounters, like you get in 'Tales of the Arabian Nights' or to a lesser extent, 'Arkham Horror' (I play RPGs, so I guess that's why). I also rather like Traitor Mechanisms, but not in every game obviously, and only when they are really well designed ('Battlestar Galactica'- yes, 'Shadows over Camelot'- less so)
Least Favourite: Diplomatic Negotiations, Pacts, Treaties etc. - the short-lived kind which is always really far open to interpretation that you make in Diplomacy, Machiavelli or even Game of Thrones. You know, you and another player make a pact to attack a third player next turn, but when you move your troops in, you notice that your 'ally' is moving a sizable portion of his troops towards your now-weakened borders. I know, that's how you win the game and totally within the spirit of things, but I just can't bring myself to do stuff like that, especially not to my mates. I prefer the treaty mechanism that 'Twilight Imperium: Rex' uses (or Dune), where it is clear when you can make a treaty, when you can break it and what either party gets out of it.
I majorly used to hate Auctions in games, by the way, but I got over that. Now I feel a bit meh about haggeling, but it's no dealbreaker for me.
And I agree, Player Elimination is terrible.
Favorite: Player Cooperation. My three favorite games, all-time, are SotM, Arkham Horror, and Dead of Winter. It might be the rpg-er in me, but I love working together towards a common goal, particularly when the game smacks you around, but you can still come around for a win. Ghost Stories is another great one, as is Pittsburgh 68 (but I'm biased on the last one; I edited it).
2nd Favorite: Unique player/character powers. Again, it's the rpg-er in me, but I want my player to be different than yours. Games that feature individual player powers really do it for me.
Least Favorite: Lack of player interaction. I hate games where what I do in the game has no bearing on anyone else, or our interactions are passive at best. Most Eurogames have this as a hallmark, particularly worker placeent games, so I tend to avoid them like the plague. Settlers of Catan, Lords of Waterdeep, Galaxy Trucker...yeah, keep those away from me.
2nd Least Favorite: "Stock Markets" in games. I find the stock market droll in real life; why would I want to play a simulation of it?
And, for the record, I actually don't mind player elimination. Ca$h and Gun$ is a fantastic game that features player elimination, as is Cutthroat Caverns. I much more deeply loathe the sorts of games where you're "effectively eliminated", such that you have 1-2 units left, but you can't actually leave, because it messes up the game dynamic for the remaining players. Sitting there, unable to affect the board or play competitively, makes me angrier than being eliminated entirely.
I completely agree with your point here; playing four separate games of solitaire isn't much fun. But Settlers of Catan isn't really the best example of that trope; there's plenty of interaction between the players. The Firefly boardgame, as much as it pains me to admit it, is a much worse offender (apart from the PvP elements of one of the expansions).
This is why I developed a deep and abiding loathing for Risk.
As for me…
Favorite: In-game storytelling. Losing because you drew the wrong card isn't much fun. Losing because your underpaid workers sabotaged your nuclear plant, causing radioactive mutants to overrun your town, is pretty fun.
Least Favorite: Slow turns. If I can make and eat a sandwich in the time it takes for my turn to come around again, you've pretty much lost me. Especially if there's no reason to pay attention during other players' turns.
I'm right with you on Risk–it simply takes so long to play that being eliminated sucks, but being effectively eliminated is even worse, because you have no impact on the game and can't leave.
You're right that Settlers of Catan isn't probably the best example of my loathed mechanic, but I've played so many games where it boils down to "I need clay. No one else has clay, because the 6 has been blocked for the last 9 turns and the only other numbers that have clay are 2 and 11. Next player…" That's mind-numbing anti-fun to me. That said, I did write a "Nuclear Catan" variant that I gladly toss in anytime someone suggests Settlers…
Favourite - Co-operative stuff, especially when one of those "Hang on a sec…you do that, and I do that, and then that happens, and then…ooooh we win! :D" kind of moments come along, everyone pitches in and stuff just works. Okay so this pretty much just applies to Sentinels (okay it's not a board game, but several other people have mentioned it ;)) from my PoV because I don't play much else, but still ;).
Least favourite - Players ganging up on each other or deliberately trying to mess each other up. I don't mind playing against someone, but I don't like a game that gets mean-spirited about it.
Once the digital version of Brass came out with asynchronous online play in became sooooooo much more fun. Waiting 30 minutes so I can play 2 cards on my turn used to be the worst. Much better when I can get on with my real life while waiting. 
Favorite: Challenge. I like a good challenge. If I want to win easily I can play a video game and shoot 8 billion zombies with my fully-automatic rocket launcher with infinite ammo. I play table games to be challenged. Also this means a challenge that involves thinking and hopefully interaction with others, not just a high chance of losing.
Least favorite: deck building games. I haven't played one yet that connected theme to gameplay in any meaningful way. They are disguised resource drafting games, I'd rather play Pit.
Bonus least favorite: Victory points. Largely intersects with the first. We're fighting off the horde of monsters bent on destroying our land in hopes that we can score more points than the other fighters, and the monsters aren't even the source of the points? WTF. Games where points are a thematic conversion of success work alright, like ticket to ride, don't love the game, but the victory points work in that game, because your goal needs to be quantified. Dominion is the worst game in the history of games.
Side note: I have played Pittsburgh 68, because I'm from Pittsburgh and I love horror movies. We were drunk, couldn't follow the rules and I'm not sure if we finished it or not. It was at magfest, and drunken table gaming is most of my weekend there.
Bonus Side note: Cash and Guns is the best drunken table gaming time I've had in a while, that is exactly how player elimination can work well.
Might I recommend Mage Knight? Admittedly, it's not a 'pure' deckbuilder, instead integrating deckbuilding into an epic RPG-like experience, but it is a fantastic game, and I find the deck-building element works really well within it.
P68 is awesome–I'm actually running a playtest of the sequel game, Roswell 51, next weekend!
And if you like Ca$h and Gun$, try out Hex Hex from Smirk and Dagger Games. As a drinking game, that one is built for a wild night!
I've heard Mage Knight does that well, and I'm not opposed to deck building as a game element when it works thematically, for example Mr. Card Game has a great little bit of deck building that is you advancing your character, it works really well in that kind of role.
I don't think I have favourites (or at least nothing springs to mind now that I'm sitting down to answer the question), it's the combination of mechanics that make (or break) a game for me.