Suppose that the order of $G$ is divisible by at least two distinct primes. Also, let $g\in G$ that order of $g$ is divisible by every prime divisor of $o(G)$ and $\forall x\in G$, $o(x)\mid o(g)$ or $o(g)\mid o(x)$.
With above conditions:
1 - What is the group $G$?
2 - Is it a cyclic group and $G=<g>$?
The group need not be cyclic, let us take $G=\mathbb{Z}_p \times \mathbb{Z}_p \times \mathbb{Z}_q$ where $p,q $ are distinct primes.
Let us take the element $g=(1,1,1)$ note that the order of this element is $pq$, all possible orders of elements in this group are $p,q,pq$, so for every element in $x \in G$ we will have $o(x) | o(g)$, but clearly this group is not cyclic since there is no element of order $p^2q$.