I suspect this is going to require more [knowledge] than any one solver has at their fingertips, so here's a partial answer that gets about half way there. (Since posting it, astute comments from user39583, Lukas Rotter and M Oehm have got us further. There are still some gaps.)
1 (card game; (_____)_) solved
This seems to be (confirmed by other evidence) some variant of (HEART)S but I don't know of one where the QS specifically scores 0. (There are variants that don't treat the QS specially at all, and ones where it scores 13, and ones where it scores 13 unless you also have all the hearts.)
2 (weird game image with French flag; 8) solved
This is from Pokemon X or Pokemon Y (I'm not sure of the exact relationship between those). I think the character in the image is called Calem; they're standing in front of the "ultimate weapon" emerging from beneath the small town of Geosenge. And it turns out (thanks to M Oehm for finding this) that in the French version of these games the town is called CROMLAC'H instead.
3 (three cards; 4 4 2) possibly solved
The game is Jass, the suit is "Rosen", the odd leftmost card ("Under") is, I believe, the same thing as the Jass (cf. English "Jack"), though curiously that name isn't actually used. If roses are the trump suit then this is the highest card in the game -- and in that case the 9 of roses, which comes next, is the second-highest. The third-highest is then the ace of roses, and despite appearances I think that's what the third card shown is. The names of these three cards are PUUR, NELL, AS and perhaps that is our 4 4 2. From the final extraction we know that the last letter of this answer must be S, which this would be consistent with.
4 (chess; 10 4) solved
This shows a draw by threefold repetition, which one could also call a REPETITION DRAW.
5 (unusual script; 6) solved
This is a Shavian transcription of a couple of lines from a section of Alexander Pope's poem The Rape of the Lock involving a game of Ombre. In that game there are three possible outcomes, best to worst: Sacada, Puesta, Codille. If I am understanding the question correctly, what we need is SACADA.
6 (card game; 3 5) solved
This turns out (thanks to user39583 for pointing it out in comments) to be a reference to a specific story involving this exact hand and these exact words and the poker player STU UNGAR.
7 (pixels; 8) solved
This is an instance of Conway's game of Life; it shows one state of a period-113 oscillator called NIHONIUM.
8 (16,000; 7 5) possibly kinda solved
The game is mahjong; on the basis of the scoring I think specifically Japanese mahjong. The tile that's being ?played? is the 8 of bamboo, called "paso". I thought this might be the fictional Japanese mahjong player SHIGERU AKAGI but OP informs me that it isn't. From the final extraction we know that the last letter of this answer must be I.
9 (four-player card game; 9 7) solved
This is a game of contract bridge; the marked bid of 2C is an instance of the convention called CHECKBACK STAYMAN.
10 (flags, rocket, checkmarks; 4 7) unsolved
The flags at the top are of England, Northern Ireland, and Wales. The flag at the left is China. From the final extraction we know that the last letter of this answer must be G. M Oehm suggested that maybe this is about a Chinese snooker player; apparently it is but not the specific one M.O. proposed. I haven't yet found one for whom I can make the numbers add up right. (As M.O. suggests, I think the ticks and crosses represents wins and losses in individual rounds of tournaments. Presumably the rocket corresponds to something like the Championship League or the World Championship.)
11 (playing card; 6) possibly solved
I think this is Pinkie Pie from "My Little Pony" playing a ... guitar, probably? It looks like a playing card but I haven't found any evidence that there actually is any game with a card like this. User yuzuo has suggested that maybe this is a PIE CARD -> PICARD, in reference to Star Trek (on account of the stars); this feels rather far-fetched to me but you never know. We also have OP's confirmation that this isn't merely PINKIE. From the final extraction we know that the last letter has to be E.
12 (woman in dress; __(__) ______) solved
This is some of the artwork from the game LO(VE) LETTER
What about the final extraction? Well, to begin with
we need 17 letters from 12 answers. The obvious way to get them is to take one letter from each answer except that for the two with more complicated enumerations we take whatever's in the parentheses. If we take the first letter from each other answer we get HEART.. RS.N.C. .VE which doesn't look promising.
But Lukas Rotter found
that there was a MLP episode called Hearths Warming Eve broadcast on that date. That fits with the "()" answers but not at all with the others. But for the ones we have guesses for, the last rather than first letter fits, as M Oehm observes in comments.
Still definitely unsolved: 8, 10, 11.
Credit where due: user39583, Lukas Rotter and M Oehm all solved some things; if you like this answer, consider going and upvoting something good of theirs as well or instead.
