Why do Sefardim place Hadas by the Huppah and Kise Eliyahu by weddings and brit milah respectively?
Like all things dealing with Sephardi minhagim, it is Kabbalistic and complicated. First the reason to have them is founded in the Zohar Helek 2, 68b, and Helek 3, 219, where it states that Hadas is a deterrent to the sitra ahra, ayyin hara and other negative spiritual forces.
Going on from there, highly mystical reasons of tikkun olam are involved as the RaShaSh elucidates quite fully on pages 22-24 of Nahar Shalom. Forgive me for not fully elucidating those reason, however they are very highly complex and there is a general issur of trying to translate such things into english, see Rav Pealim 1, YD 56.
In Bigdei Kodesh from Rav Shimon Dror p 144, and also in A treasury of Sephardic Laws and customs by Rabbi Herbert Dobrinski 15-16 a number of midrashic sources are brought as well. Such as Bereishit Rabba Lekh Lekha parasha 47, dealing with the Brit of Avraham Avinu. As the other works are not online and I don't have time to hunt up their bibliographic sources, I will quote from R' Dobrinski's work:
Their is a view that this also occurs, not because of giluy Eliyahu HaNavi, but simply because of the kedusha of the event, and the same holds true of weddings.