So Perrin knows Faile needs help. He zips around the world of dreams until he finds a falcon and then he concludes she is there in the real world. Why? How? This doesn’t make any sense.

my only guess is that Perrin used a God power  previously undescribed, he just couldn’t admit to himself he had it so he made believe he found her the world of dreams. 

Its actually the 2nd time Perrin rescued Faile this way, the first time Faile had triggered a trap which put her into a Coma, in a much earlier book(TBR in Tear i think) and he had to go look for a Falcon in the Wolf Dream to resuscitate her. 
Don't forget that Faile means Falcon in the old tongue.

Yeah, but in that one a terangreal stole Faile’s soul and stuffed it into the world of dreams in a semipermanent way. She would have died if she wasn’t rescued and her soul released from the trap inside the world of dreams.

At the end of the last battle, Faile had no Terangreal, no access to the world of dreams and even had she fallen asleep, and her dream brushed against the world of dreams, it should have been her, not a falcon. 

Need.  Just as the Aiel Wise Ones would use Need in Tel'aran'Rhiod to find sources of water when they needed to found new holds in the Waste, and how Nynaeve and Elayne used it to find the Sad Bracelets and the Bowl of the Winds.  They didn't know where the things they needed were, or even what they were in the latter case, but Tel'aran'Rhiod brought them to what they needed nonetheless.  As for why Faile appears to Perrin as a falcon in the Dream, it was the Wolf Dream's representation of her for Perrin, not necessarily her herself.

Edited June 23, 20206 yr by Thrasymachus

Very good explanation.