No, it is not necessary.
are all odd numbers prime?
No.
There are exactly as many odd numbers as primes (Euclid, Cantor), and exactly one even prime (namely 2), so there must be exactly one odd nonprime (namely 1). Cosmologist: 3 is prime, yes it is true.... ... 10 is prime, 11 is prime, 101 is prime...
How to prove that all odd numbers are prime
Yes.
And one even number is prime---see Rooster's answer.
https://www.reference.com/math/prime-numbers-odd-5f0c2d756b96b334
No, it's not necessary. The number only divisible by 1 and itself is prime number.