There is a misconception nowadays about Sephardim. Sephardim means those who are from Spain. Iraq is not spain, nor is Syria and nor is Iran. Jews originating from these countries are not Sephardim rather they are mizrachim. The general uniformity you find amongst the “Sephardic” world comes as a result of the influence which the shulchan aruch had on the jewish world and the kehillas who adopted the psakim of the shulchan aruch. In 1992, Rav ovadia Yosef ztz”l when he was chief rabbi attended an event in Spain commemorating 500 years since the inquisition. At this event the king of Spain asked Rav ovadia why so many call themselves Sephardi when in essence they are not of Sephardic origin. Rav ovadia replies that it was due to the acceptance of the shulchan aruch that they called themselves sephardic.
Of course Jews escaping the inquisition reached Syria, Iraq, Persia etc but these were few in number. The vast majority of spanish Jews fled to morocco, Italy, Holland and the Balkans/greece/turkey. Minhagim amongst these communities are similar as is the nusach. The nusach of real Sephardim is slightly different to the nusach of the “Sephardic” siddurim available today.
With regard to pesach, a lot depends on family minhagim and the towns they were from. Stringencies developed based on maasim which happened in that town. For example some don’t eat sugar because once they found a crumb in the sugar left over by the vendor, rice wasn’t eaten because they would be shipped in the same bags as flour, or they would bleach the rice with flour. Some don’t eat chickpeas because in arabic Hamus sounds like chametz…. and so on.
The pasuk in ovadia says vegalut yerushalayim Asher bisfarad. There’s a mesorah that Jews were living in Spain since the times of the bayit rishon as Shlomo hamelech sent Tax collectors there (one is buried in tarragon). It is also claimed that Toledo was built by jewish expulsees when the first bet hamikdash was destroyed. This is documented by the Abarbanel who says his family as in Spain for 2000 years (they were descendants of david hamelech)