Standard factoring problem $\Pi_1$ is 'Given integers $N$ and $M$ is there a factor $d\in[1,M]$ of $N$?'. This is in $NP$ since such a factor is the witness and in $coNP$ since one can check all the prime factors of $N$ are bigger than $M$. So the standard factoring problem $\Pi_1$ is in $NP\cap coNP$.

Factoring problem $\Pi_2$ 'Given integers $N,L,U$ is there a factor $d\in[L,U]$ of $N$?' is $NP$-complete under Cramer's conjecture of prime gaps. So factoring problem $\Pi_2$ cannot be in $coNP$ under standard conjectures.

Factoring problem variant $\Pi_3$ 'Given integers $N$ is there $d_1,d_2\in\mathbb Z$ such that $$\sqrt N<\gamma(d_2-d_1),\quad d_1<d_2<\delta d_1\quad\mbox{ and }\quad d_1d_2=N$$ holds where $1<\gamma,\delta$ are constants?' is clearly in $NP$ and it generalizes $RSA$ type factoring problem as we will explain shortly. However it is neither clear if $\Pi_3$ is in $coNP$ nor clear if $\Pi_3$ is $NP$ complete problem.

- Is $\Pi_3$ in $coNP$?

If $\Pi_1\in P$ then $\Pi_2$ could still be not in $P$ since it is $NP$ complete under a believable conjecture.

- If $\Pi_1\in P$ does it help decide $\Pi_3$?

Note that if $d_1,d_2$ are prime then $\Pi_3$ includes some $RSA$ type problems and if $d_1,d_2$ are prime then $\Pi_3$ is in $coNP$ and if $\Pi_1\in P$ then $\Pi_3\in P$ holds.