According to the Gemara and Shulchan Aruch, a person in the Diaspora is required to live in his sukkah also on Shemini Atzeres. In practice, many people do not act this way. Both “Shaarim Metzuyanim Behalachah” (138:3) and “Piskei Teshuvos” (668:1) write on the question and bring many sources which justify this.

The recital of hallel on Rosh Chodesh is a minhag and so Sefardim don’t say a berachah over it. Another minhag is the lighting of Chanukah candles in shul with a berachah, and the Mishnah Berurah (671:44) specifically compares these two minhagim with regards to berachos. However in this latter case, Sefardim do so a berachah. Why the contradiction?! The Aruch Hashulchan (Orach Chaim 671:26) discusses this question stating that there are visitors and unmarried people who don’t have a house and thus fulfill this mitzvah in shul, and even if this is not the case, then the main pirsumei nisa is today in the shul and thus one should say the berachah there.