Project G came first, failed, and Gast moved on to Project S—Genesis is older than Seph by at least a year. EC has stated that Angeal is also older than Seph, which is a retcon from the timeline established in CC, but gives us a timeframe nonetheless. Gast probably did some traveling back and forth between the two locations during the overlapping end of G and beginning of S, leaving the majority of breakdown and setup (respectively) to his research assistants on the two projects. This is what research assistants are for, to be fair, and the differing methods between the two wings of the project were conceptualized by those same assistants—Project G was Gillian's brainchild and Project S was Hojo's, with Hollander and Lucrecia serving as secondaries.
My theory on Genesis' parentage is that he was the result of more careful breeding than the other two, which was seen as maybe being part of the issue. We know he got genes from Gillian (not cells, the word used is analogous with "genes" even in Japanese), but she's not technically his mother; Hollander indicates that this extra degree of separation was assumed to be why he was born "normal" and thus dubbed a failure, while Angeal is Gillian's biological son and is the closest thing to a true duplicate of Jenova itself that anyone has ever seen.
Based on real-world scientific methods, Genesis was probably originally produced in a test tube and then transplanted into a host mother to gestate; my headcanon has always been that said host mother was the same person who gave birth to Weiss and Nero, given that Usher refers to her in Dirge PlayOnline as having been "a fantastic producer," which I don't think would really be appropriate to say of someone who successfully had a total of two science project babies. My Genesis' biological father is Veld of the Turks, although he never finds out and wasn't actually personally involved—samples were taken during a "routine physical examination" so Gast that could use him as breeding stock. (My headcanon also says this was the same thing he intended to do with Vincent, but things didn't end up going that way because What The Fuck Lucrecia.)
As for Project S, we don't actually know what the S stands for! However, going by the previous Jenova Project being Project Gillian, with Gillian serving as the lead assistant, we can assume this is the first initial of Hojo's given name.