Is there a number P such that every sufficiently large integer can be written as the sum of at most P different primes?
Yes, it's the Vinogradov's theorem :
But it seems to be true for any number > 7 : Every odd number >5 is the sum of 3 primes (proved in 2013) http://arxiv.org/pdf/1305.2897v1.pdf
Yes. Vinogradov proved that every large odd integer can be written as the sum of three primes; the proof actually gave an asymptotic formula for the number of such representations, which is easily seen to be larger than the number of representations with repeated primes (that is, in the form $p_1+2p_2$ or $3p_3$) or the number of representations where one prime equals $3$. Therefore, all large odd integers can be written as the sum of three distinct primes greater than $3$. Trivially then, all large even integers can be written as the sum of four distinct primes (one of which we can take to be $3$).