Since 2 is the only even prime, we get the first counterexample when we have two composite odd numbers in a row... namely, 27.

For the second question just take 2*3*5 as counterexample... or 2*2*2 for that matter.

EDIT: Actually the first counterexample for the "odd" problem is 11, I wasn't thinking clearly when I wrote 27. (27 is the smallest odd number greater than 4 that cannot be written as the sum ofat most2 primes.)