Good observations, so I'll focus on the basics.
Jews don't explain the Talmud much because neither they nor the Gentiles seem much interested in the Gentiles learning it. So they take lots of things for granted. As I said, to be sure you've got everything called "Talmud", you need Mishna, plus Bavli, plus Yerushalmi (the last two are both called Gemaras, and are synonymous with Babylon and Jerusalem). People who object to Jewish teaching often throw in excessively many other sources that are indeed Jewish but are not core and are often quite recent, so those get filtered too.
When you see a cite with a or b, the code is that that means it's a page of the Talmud Bavli (also including Mishna) as printed by Bomberg. When you see two numbers such as with a colon, that means it's a section or long paragraph of the Talmud Yerushalmi (also including Mishna). There are other ways to refer to them, such as by chapter, which is further confusing. So in this case you got distracted by starting with Yerushalmi when the main cites you want are in the (larger) https://www.sefaria.org/texts/Talmud/Bavli text. Clicking Sanhedrin there will immediately show you 59a is in chapter 7 of the Bavli Sanhedrin. The Yerushalmi would have a parallel passage, but honestly I don't know who has published a reference of parallel passages between the two Talmuds (Gemaras), and usually people don't care because they're only citing one of the two and they expect you to know that code that tells which one.
[Copilot is accurately saying there are 13 sections in chapter 7 (7:1 to 7:13) in Yerushalmi, but is getting the sections wrong. You can find these in Bavli by hand by searching for the word "mishna:" and comparing them to Yerushalmi, which is rather clunky. So to compare 59a, I scroll up and down and find that 55b has a mishna about blaspheming and 60b about idolatry (long section, 10 pages or 5 folios). I compare the mishna headers of Yerushalmi and find that these are 7:8 and 7:9, so 55b-60a Bavli corresponds to 7:8 Yerushalmi. But the parallels often diverge greatly: so the 59a quote is not in 7:8, only in 59a.]
TLDR: The first link in each paragraph of my rebuttal refers to the online page where the citation is best understood to refer (but not to subparagraph number). A few of them, having come from 19th-century texts that are not online, cannot be double-checked because the original source (sometimes August Rohling) misspelled them and/or didn't provide enough data for a valid citation.
However, since I didn't include subparagraph number, it would be appropriate for me to go back and do that. (Using both a-b and colon means paragraph in Bavli.) 1. Sanhedrin 59a, quote not found. 2. Avodah Zarah 26b, quote not found. 3. Sanhedrin 59a:2. 4-5. "Libbre David 37", source not found. 6. Yevamot 11b, quote not found. 7. "Schabouth Hag. 6d", source not found. 8-9. Mishneh Torah, Foreign Worship and Customs of the Nations 10:1. 10. Tur, Choshen Mishpat 388, quote not found. 11. Tur, Choshen Mishpat 266:1, quote not found. 12. "Szaaloth-Utszabot, The Book of Jore Dia 17", source not found. 13. Baba Metzia 114b:2. 14. Yalkut Shimoni on Nach. 499:2. 15. Midrash Talpiot 315:1. 16. Avodah Zarah 37a:1. 17. Mishneh Torah, Kings and Wars 8:2. 18. Tosefta Avodah Zarah 8, quote not found. 19. Tur, Choshen Mishpat 388, quote not found. 20. Tur, Choshen Mishpat 348, quote not found.
Among those where the anti-Talmud transcribers gave bad citations, #4-5 has six offline candidates that it could be, #7 has one, #12 has three. (Add: I finally found #15.) In a couple cases I found the correct more common name intended by the given source name. Besides those 4 offline, there are 8 online where the quote is findable and 8 online where (my longer link shows) the quote came from elsewhere, such as a later nonbinding commentary on the passage quoted. If you need more help interpreting my own brief takes, please let me know.
In short, it's likelier that this crazy misquotation meme is being circulated by Jews trying to get us to learn Talmud for ourselves by their self-deprecation than being circulated by Gentiles who are such horrendous researchers as to believe it accurately reflects Talmud.
Good observations, so I'll focus on the basics.
Jews don't explain the Talmud much because neither they nor the Gentiles seem much interested in the Gentiles learning it. So they take lots of things for granted. As I said, to be sure you've got everything called "Talmud", you need Mishna, plus Bavli, plus Yerushalmi (the last two are both called Gemaras, and are synonymous with Babylon and Jerusalem). People who object to Jewish teaching often throw in excessively many other sources that are indeed Jewish but are not core and are often quite recent, so those get filtered too.
When you see a cite with a or b, the code is that that means it's a page of the Talmud Bavli (also including Mishna) as printed by Bomberg. When you see two numbers such as with a colon, that means it's a section or long paragraph of the Talmud Yerushalmi (also including Mishna). There are other ways to refer to them, such as by chapter, which is further confusing. So in this case you got distracted by starting with Yerushalmi when the main cites you want are in the (larger) https://www.sefaria.org/texts/Talmud/Bavli text. Clicking Sanhedrin there will immediately show you 59a is in chapter 7 of the Bavli Sanhedrin. The Yerushalmi would have a parallel passage, but honestly I don't know who has published a reference of parallel passages between the two Talmuds (Gemaras), and usually people don't care because they're only citing one of the two and they expect you to know that code that tells which one.
[Copilot is accurately saying there are 13 sections in chapter 7 (7:1 to 7:13) in Yerushalmi, but is getting the sections wrong. You can find these in Bavli by hand by searching for the word "mishna:" and comparing them to Yerushalmi, which is rather clunky. So to compare 59a, I scroll up and down and find that 55b has a mishna about blaspheming and 60b about idolatry (long section, 10 pages or 5 folios). I compare the mishna headers of Yerushalmi and find that these are 7:8 and 7:9, so 55b-60a Bavli corresponds to 7:8 Yerushalmi. But the parallels often diverge greatly: so the 59a quote is not in 7:8, only in 59a.]
TLDR: The first link in each paragraph of my rebuttal refers to the online page where the citation is best understood to refer (but not to subparagraph number). A few of them, having come from 19th-century texts that are not online, cannot be double-checked because the original source (sometimes August Rohling) misspelled them and/or didn't provide enough data for a valid citation.
However, since I didn't include subparagraph number, it would be appropriate for me to go back and do that. (Using both a-b and colon means paragraph in Bavli.) 1. Sanhedrin 59a, quote not found. 2. Avodah Zarah 26b, quote not found. 3. Sanhedrin 59a:2. 4-5. "Libbre David 37", source not found. 6. Yevamot 11b, quote not found. 7. "Schabouth Hag. 6d", source not found. 8-9. Mishneh Torah, Foreign Worship and Customs of the Nations 10:1. 10. Tur, Choshen Mishpat 388, quote not found. 11. Tur, Choshen Mishpat 266:1, quote not found. 12. "Szaaloth-Utszabot, The Book of Jore Dia 17", source not found. 13. Baba Metzia 114b:2. 14. Yalkut Shimoni on Nach. 499:2. 15. "Midrash Talpiot 225", source not found. 16. Avodah Zarah 37a:1. 17. Mishneh Torah, Kings and Wars 8:2. 18. Tosefta Avodah Zarah 8, quote not found. 19. Tur, Choshen Mishpat 388, quote not found. 20. Tur, Choshen Mishpat 348, quote not found.
Among those where the anti-Talmud transcribers gave bad citations, #4-5 has six offline candidates that it could be, #7 has one, #12 has three, and #15 has one offline edition but another unsearchable Hebrew edition with different pagination is available. In a couple cases I found the correct more common name intended by the given source name. Besides those 5 offline, there are 7 online where the quote is findable and 8 online where (my longer link shows) the quote came from elsewhere, such as a later nonbinding commentary on the passage quoted. If you need more help interpreting my own brief takes, please let me know.
In short, it's likelier that this crazy misquotation meme is being circulated by Jews trying to get us to learn Talmud for ourselves by their self-deprecation than being circulated by Gentiles who are such horrendous researchers as to believe it accurately reflects Talmud.