Cocoon was made by the Fal'Cie. That is mentioned in the game and by that Pulse Fal'Cie data log entry in History & Society. The analects must all be taken with a pinch of salt because they are written by individuals who can distort facts.

Barthandelus never mentions Lindzei. He only refers to The Maker. Therefore it is unclear whether there is actually a God called Lindzei. It is also unclear whether there is a Goddess called Pulse. Lindzei is also never referred to anywhere on Cocoon. It would be more likely that Lindzei (who is basically Lucifer) was an idea created by the people on Pulse to further fuel their hatred of Cocoon.

The Pulse Fal'Cie looking for a portal or key to finding The Maker is perhaps unrealistic. But then again, we don't know what the purposes of these Fal'Cie are. Perhaps the Pulse Fal'Cie were given the purposes of terraforming and therefore they can only explore, whereas the Cocoon Fal'Cie were given the tasks of looking after humanity, and therefore can breed them.

The analect you refer to talks about the Fifth Ark. But the party, while there, believe that Barthandelus deliberately brought up the Fifth Ark to Cocoon in case he would ever need it. Cocoon is built from material from Pulse, so it is fairly easy for them to steal stuff.