Is it possible to generate a secure permutation F over 32-bit integers even if F(0) ... F(n) is public knowledge?
Yes, it is possible to implement the primitive asked, with a 32-bit block cipher that is secure (indistinguishable from a random permutation) no matter how many input-output pairs are known, keyed with a fixed secret randomly-chosen key. That's a standard building block in Format Preserving Encryption.

One such block cipher is: Louis Granboulan and Thomas Pornin, Perfect block ciphers with small blocks (in proceedings of FSE 2007).

Emil Stefanov and Elaine Shi's FastPRP: Fast Pseudo-Random Permutations for Small Domains (Cryptology ePrint Report 2012/254) claim a speed improvement, and review some earlier work.

Update: Ricky Demer kindly pointed another recent paper: Ben Morris and Phillip Rogaway, Sometimes-Recurse Shuffle, Almost-Random Permutations in Logarithmic Expected Time (Cryptology ePrint Report 2013/560).