SPI security of FE1 scheme
why we are allowed to replace PRF with random function in security proof?


That's just how security proofs for these kinds of Feistel schemes work - you say something about the cipher when the round function is a truly random function, then you argue that because a PRF is indistinguishable from a random function for a computational adversary, the security proof still holds when your round function is a PRF. I think they use this technique in the original Luby-Rackoff paper, too.