See here for sources:
1) The Mishna itself is not clear about Rabbi Shimon's reason. Rashi suggests that he holds like Rabbi Meir that it belongs to the owner of the upper garden, but that he is mafkir towards the owner of the lower garden, because of embarrassment?
Yet he is kicking and screaming in court for rights to it? Maybe not, we are just establishing what the halacha is, and people will follow it. Ramban has the opposite, that it belongs to the lower owner (who has air rights), who is mochel / mafkir the top part.
It could be that it is neither. Rather, Rabbi Shimon holds it is like a dofen akuma, that the upper garden bends, 10 tefachim or as far as one can reach without effort, so the side soil is like the topsoil of his garden. That it is like a pavilion (apiryon) which bends (namtaya).
2) See source , a comparison of Bavli and Yerushalmi. According to Bavli, everyone holds like Rabbi Shimon. According to Yerushalmi, what Ephraim reports in the name of Resh Lakish is that we split. Presumably because we don't decide between Rabbi Meir and Rabbi Yehuda. See next note about teku of mamon where you split.
What is reported in Bavli in the name of the academy of Rabbi Yannai is what Rabbi Yochanan says in Yerushalmi in the name of Rabbi Yannai. Meanwhile, in Yerushalmi, the academy of Rabbi Yannai have a different position, that Rabbi Shimon extends to 10 tefachim. It is simplest to read this as a dispute, but perhaps these can be read as complementary.
3) See source , the Rosh. Regarding the teku, this is classing Rabbi Yirmeyah, who was eventually thrown out of the bet midrash for asking about a bird with one leg in bounds and one leg out of bounds.
The Hagaha (pictured) notes that Rabbenu Chananel says that in case of teku of mamon, the rule is to split.
Note that of the two cases we have in our gemara and Rashi which results in teku, the Rosh (pictured) only has reaching the protrusion and not the root. It seems that he, as well as the Rambam, have a different girsa which doesn't have the case of reaching the root but not the protrusions. Interestingly, Rosh goes on to argue on the Teku, saying that, based on the **unspoken** reason for Rabbi Shimon of the upper garden owner being mafkir, here he would not be mafkir, because he can get at it from the protrusions.
4) Rashi defines the yerek of the Mishna as onions or garlic. Garlic is of the onion family. Both have the bulbs under the dirt. I am not sure why Rashi wants to establish the case in this manner, particularly since in the gemara, Rava makes the entire dispute about to protrusions.
5) King Shapur praises Rabbi Shimon. Rashi maintains this is the actual King Shapur of Persia. (And Maharatz Chayes says that since establishing law is one of the seven Noachide commandments, this is part of the Torah which one may teach to non-Jews.) The opposing position is that this is really Shmuel, who the Amoraim sometimes called King Shapur. I can understand it as a nickname, since Shmuel was close to king Shapur and the Sasanian government.
It is unclear if this was really a nickname of Shmuel. See Pesachim 54, one such instance. Rava says "I will tell you something even King Shapur doesn't say", which is an idiomatic boast. The setama degemara interjects that this is a reference to Shmuel, but this could be analysis outside of the social context in which the statement was made. Interestingly, there is another version which follows in that gemara, that some say Rav Pappa made the statement, in which case King Shapur refers to Rava.
6) I haven't seen anyone say this, but it is **important** to note that King Shapur is making a pun. That is, upon hearing a statement of **Ephraim** who is **noteh** like Rabbi **Shimon**, he says that his Apiryon (=Ephraim) extends (nimtaya) to Rabbi Shimon.
This pun is important because, unlike one rejected hava amina in Rashi, this is not addressing Rabbi Shimon of the Mishna directly. It is addressing specifically the statement of Ephraim in the name of Resh Lakish.
7) See source . Given that King Shapur makes this statement, we might expect to see a dry 55-page article about Sasanian attitudes towards adjacent upper and lower gardens. But Shai Secunda in "The Iranian Talmud: Reading the Bavli in Its Sasanian Context" says there is nothing directly in Sasanian law.
Maybe we can relate it to Sasanian law about fields adjacent to a riverbed geting some portion of the riverbed (to the ear), especially if we are dealing with a dofen akuma conceptualization.
8) Apiryan as grace is apparently a Middle Persian word. There are two ways of reading the word. Rashi has apiryan, grace. Rabbenu Chananel has apiryah, like pru ureva, that rabbis such as Rabbi Shimon should increase.
See source . If we read it like apiryon (with a cholam as in our printed texts, rather than with two yuds), then the word is found in Shir HaShirim perek 3 as a hapax and probable foreign loan word.
If we would like to be mystical, then stam Rabbi Shimon in the Mishna is Rabbi Shimon ben Yochai. And in the Zohar parashas Terumah, the pasuk in Shir Hashirim about the Apiryon is taken to refer to the Bet HaMikdash, functioning as an apiryon connection between the upper and lower. As in the English explanation I copied (which is expansive rather than a literal translation), the connection is between the Upper Garden (of Eden) and the Lower Garden (of Eden).
So while Rav Saadia Gaon doesn't know Zohar, King Shapur does, and says that according to Rabbi Shimon, there is an apiryon which extends from the upper to lower garden.
9) See the variant texts in source . Ephraim Mekoshaah is an error, as that would refer to the student of Rabbi Meir, not the student of Resh Lakish.
Many texts are expansive of what King Shapur says. See inside.
Also, many texts have Rabbi Shimon ben Lakish explicitly. I think the correct would be the ambiguous Rabbi Shimon, though these texts would reflect how the statement was commonly, and perhaps correctly, understood.