Read Milnor Stasheff: Characteristic classes. For the first question: The total normal Stiefel Whitney class of $M$ is $(1+w_1(M))$, hence the inverse is $W(M) = 1 + w_1(M) + w_1(M)^2 + ...$
For the second Question one uses that the tandent bundle $\tau$ of $RP^n$ satisfies the equality $\tau \oplus \epsilon$ is isomorphic to $n+1$ times $\gamma$, the tautological bundle. Hence $w_i(RP^n)$ the binomial coefficient choose $i$ from $n+1$ Put the two together you get the answer to the second question