Officially, that expedition was made by the Commitee of Inquiry about the Second Impact. This expedition should investigate about the causes of Second Impact (in fact, after, they would invent that an asteroid hit the Earth causing that disaster). Fuyutsuki was chosen by the Seele as civilian member, because the ONU didn't want to let the Seele make that expedition by itself.
Maybe, Fuyutsuki was chosen by Gendou, who wants to meet him again, now that his studies has become interesting for the Seele's projects.
Officially is for this. But we can suppose a lot of things about it, for example that the Seele wants to resecue the Adam's embrion, because there was no survivors from the Katsuragi expedition, except Yui & Gendou (who leave before the disaster) and Misato (who...well I think she didn't give a damn about Adam XD). So the embrion was still there. The Lancea Longini was too cumbersome to be resecued at that time.
Well, actually I don't have a Red Cross Book in my hands, so I can't say you who wrote it, but I read the translation (in italian, sorry

) in a website and it seems that in this book there are the schedule of all the characters and a glossary of all the terms used in NGE.
However, this pamphlet is officially approved as canon material by Anno and GAiNAX, so...it should be a well-done one ^^