This is what i have done

Let f(x)=x^(n-1)-1

there exist x such that gcd(x,p)=1, just let x=1 and x^(n-1)=1 mod p.

f'(1)=/=0 since it's still 1. so there exist a solution in Zp that f(x)=0

And part 2, since with part 1, Zp has all (p-1)th root, it's obvious that if p=5, Zp had all 4th roots, one of them is a primitive root. But that answer is too stupid