I'm not familiar with "Darling in the Franxx" at all, but I think the solution to your problem is the exact same one I used for my Danganronpa RP.

The first thing you want to do is talk to some fans of the show and see why people are watching it. You want to know why people are watching so that you can make changes and be confident they won't drive people away. When I talked to people about Danganronpa, it seemed like everyone was more interested in the exhilarating "solve the murderer or die" mechanic than the individual characters and story. This allowed me to confidently change the origin of the characters. Instead of all the characters being affiliated with hope's peak, they are affiliated with an original organisation. The mascot villain is a centerpiece for the franchise however, and I worked hard to replicate his behavior as best as I could. also, Danganronpa is traditionally split 50/50 even as far as character sexes go, and I just let people make who they wanted to make. If you're going to do crossover shenanigans, my advice is to make an original RP and just say it was inspired by "Darling in the Franxx" and "Evangelion." This gives you and your players the most freedom, and will lure in any fans of those shows who are brave enough to click on your thread. 

As for how to make a plot or if you should share the work with a co-GM, there is no right answer, and I could spend all day talking about the topic. You sound like you want to combined ideas from a lot of different shows, which makes me thing you'd have an easier time with an original setting inspired by these other shows. I have never used a Co-GM, but I personally feel that making the big monsters. There's also no rule saying that you have to pick a particular method. You could have all the players fighting their own monsters, and then decide to have one big monster show up. Of course, you need to have good communication with your players so that the big monster can arrive at a time everyone's able to fight it.