Calculate number of required pairs for differential cryptanalysis
See Heys' Tutorial on Linear and Differential Cryptanalysis (google it) for a gentle introduction. 

Ali Aydin Selcuk has carefully analyzed this success probability in a number of papers.

For differential cryptanalysis roughly $c/p$ plaintext ciphertext pairs are needed where $c$ is a small constant dependent on the cipher.

As for your second question, unless $p$ is much larger than about $2^{-80}$ there is no practical attack. After all, you need to encrypt at least as many as $c/p$ plaintext ciphertext pairs to collect data before you can mount the attack and the quantity $c/p$ is a lower bound on at least your time complexity (possibly also your memory complexity depending on your attack).

However a $p$ value that is appreciably larger than $2^{-k}$ where $k$ is the keylength of the block cipher indicates a weakness in that the cipher is weaker than expected, as in Biham and Shamir's attack on DES.