At first glance this appears to meet all the requirements. Let me start with some prose comments, none of them are important enough to make this nomination fail, they're just suggestions we can discuss:
having won the Women's World Rapid Chess Championship once and the Women's World Blitz Chess Championship twice --> perhaps in brackets add the years
taught her and her younger sister Mariya the game when they were each two years old --> this made me think they were twins, but after a quick check that appears not to be the case. Perhaps a rewrite?
Rephrased to: "Muzychuk grew up in a chess family where her younger sister Mariya is also a Grandmaster and her parents work as chess coaches, having taught her the game from when she was two years old." Sportsfan77777 (talk) 13:51, 5 January 2022 (UTC)[reply]
Muzychuk's sister Mariya was the 2015 Women's World Champion in classical chess. --> this would fit better at the end of the previous paragraph I think
one thing that may need a bit of extra attention I think is the fact that she sometimes played in women only and sometimes in mixed tournaments. This is not possible in many other sports, so perhaps the first time she won an open tournament add that it was open to all players regardless of gender?
Hmm, I'm not sure she has won any key open tournaments. One thing to note is that most tournaments are open. For the most part, only national, continental, and world championships are women-only. You start out playing mostly open tournaments except for the youth levels of those. Then, if you get to the highest level, you end up playing more women-only tournaments because you participate in all of the championships and there is also the Grand Prix and the Candidates to help decide who participates in the World Championship. Sportsfan77777 (talk) 18:10, 12 January 2022 (UTC)[reply]
At a younger age, she may not have particularly high finishes at open tournaments because she usually played above her level (as most young players do). At an older age, she would still probably play above her level most of the time. Sportsfan77777 (talk) 18:10, 12 January 2022 (UTC)[reply]
Some of her better results at open tournaments are: the InventiChess GM norm in 2009, Wijk aan Zee Group B in 2010, Wijk aan Zee Challengers in 2014, and Gibraltar in 2016. (She didn't win any of those, but just had good performances.) Do you think that's clear? Sportsfan77777 (talk) 18:10, 12 January 2022 (UTC)[reply]
better in the rapid event than the blitz event, finishing in fourth place --> perhaps add where she finished in the blitz
I don't think the exact result is so important if she didn't come close to medalling. I added a source for the blitz event though so it's there if someone is interested (and supports the "better in the rapid event" part). Sportsfan77777 (talk) 18:10, 12 January 2022 (UTC)[reply]
AVS Krasnoturinsk, who won team bronze and silver medals in her two appearances as well as an individual silver in her first appearance --> this doesn't read very well, perhaps break up with some punctuation after two appearances?
Fixed: "She started out playing for AVS Krasnoturinsk, winning team bronze and silver medals in her two appearances as well as an individual silver in her first appearance." Sportsfan77777 (talk) 13:51, 5 January 2022 (UTC)[reply]
Muzychuk has played perennial contenders CE de Monte Carlo --> played for
likes shopping and doing things outside of home --> not so sure if this is really encyclopedia material. It would be more interesting if there was anything about how the sisters compared in their chess styles, and their competitive records facing each other.
I wasn't sure about inclduing a whole annotated game, but the only chess article that is at FA level has one (see George H. D. Gossip), so I guess it is fine. I guess this is still considered brief, as required per MOS:QUOTE. One thing that George H. D. Gossip does better is the use of bold text for the actual moves. Makes it easier to read and is standard in chess reviews.
Similar types of things have been brought up before. People have different opinions on the section in general. I don't think it's likely for a discussion to end with any sort of agreement. Sportsfan77777 (talk) 18:10, 12 January 2022 (UTC)[reply]
sources 47 and 48 do not fully cover the claim "During the event, she became the fourth woman to reach a rating of 2600 after Judit Polgár, Koneru, and Hou Yifan". 47 says "is poised to become". Source 72 mentions she is the 4th woman to pass 2600.
source 49 didn't work for me. I managed to download a zip file from archive.org but it failed to open
That's what I did. I was able to get it to open. (I think the file itself is still available on the FIDE website today, although I'm not sure it's linked anywhere. FIDE is not the most organized compared to other sports.) Sportsfan77777 (talk) 13:51, 5 January 2022 (UTC)[reply]
At the end of 2016, Muzychuk won both the Women's World Rapid Championship and the Women's World Blitz Championship in Doha, thereby also defending the latter of these two titles from 2014. She became the third player to win both events in the same year after Susan Polgar in 1992 and Magnus Carlsen in 2014. --> source 4 seems ok, but ideally there are written sources for winning the championships, sources 70 and 71 will do.
I intended the last two sources to support the previous sentence(s). I could repeat them if that's not clear. The video is only used for the statement about the previous rapid and blitz winners. Sportsfan77777 (talk) 18:10, 12 January 2022 (UTC)[reply]
I recommend you archive url's. I had a real issue with my Manon Melis FA when the Swedish Football Association website had a tech problem and a dozen urls stopped working. It took me 2 days to recover.
I think the article could benefit from an extra Honours section, just like the football players have, see Thierry Henry. She has won so much, would be easier for the reader if it is grouped altogether.
I like that the football project seems to have an established format for these sections. With tennis, I've run into the issue that because we don't have clear guidelines, people will just add anything to these sections, particularly with players who are still active. I think there would be a similar issue here (sort of like the Russian version of this article). I also don't think any chess players really have this sort of section (for Magnus Carlsen, it is a list of awards, not so much tournament wins or medals). I don't really have a good idea for a format for starting it. I did try to put all of her key accomplishments in the lead. Sportsfan77777 (talk) 18:10, 12 January 2022 (UTC)[reply]