Rosh Hashana Daf 4

1. Tosfos 4b sv Pz"r - Rach and Rabbeinu Tam hold Shmini Atzeres is another Regel to stop aveilus.  We don't pasken like this, but Aruch Hashulchan brings it down l'tzorech.  Could be that if a) if Sukkos stopped Aveilus, Shmini Atzers can't either, or b) you need some chol before the regel for the regel to interrupt aveilus, and there is nor chol between Sukkos and Shmini Atzeres.  A possible Nafka mina would be if the mes died on Sukkos - there is no chol, but you're not  "double dipping " with both Sukkos and Shmini Atzeres.

2.  Shmini Atzeres separate for Bracha - Rabbeinu Tam held that it is not called Sukkos in davening.  This would work better according to the Torah Temima that on Yom Tov we don't say "mekadesh yisrael vehazmanim," rather "mekadesh Yisrael v'yom hasukkos" or "v'yom HaPesach."  Rav Yosh Ber vehemently argues on him (Nefesh Harav p. 182 and here).

This Tosfos slightly supports Torah Temima, but, would depend on the following question: a bracha's chasima must be like its content, so the fact that you said Chag Hasukkos in Yaaleh v'yavo and mekadesh yisrzel v'yom hazikaron in chasima is not a stirah.  What is the halcha is a) you said chag hamatzos in Yaaleh v'yavo on Sukkos, or b) didn't say any holiday name in yaaleh v'yavo and just said something like "byom tov hazeh", if you were yotzei shmone esray.  (Grid in Harirei Kedem I Siman 151 has a possibility saying Sukkos on Shmini Atzeres works but not any other mix-ups.)

Also this should depend on the Halacha mentioned in Shulchan Aruch Orach Chaim Siman 268:4 - if you mentioned Shabbos in Shmone Esra yeven though you didn't make mekadesh Hashabbos you are yoztei. Aruch Hashulchan 268:9-10 vehemently argues on this Beis Yosef because Chazal made a bracha for Shabbos, and if you mentioned shabbos but not the Bracha you shouldn't be yoztei.  I'd assume same  din, whether you hold like Beis Yosef or argeue on him, should hold for Yom Tov.  But, to answer the question of Aruch Hashulchan, we can say this Tosfos is the key: true you need to say mekadesh yisrael vehazmanim on Yom Tov, but the key ingredient is mentioning the name of the holiday in the middle of the Bracha.  (With this we can push off Torah Temima also and say mentioning holiday in the middle of the bracha is enough.)  I do admit that I don't like the logic of Tosfos, that mentioning the holiday within the bracha without a chasima of the holiday (bidieved) is yotzei.

3. Mosar Pesach.  See Gra"m and R' Aharon Lichtenstein, Shiurei Hare"l to Pesachim p. 217