We know that computing number of prime factors implies efficient factoring algorithm (How hard is it to compute the number of prime factors of a given integer?).

Let $\omega(n)$ be number of distinct prime factors of $n$. Knowledge of below three do not seem to give a factoring algorithm.

Square-freeness (already have efficient square-freeness tests for univariate polynomials).

Number of bits (MSB) and parity (LSB) of $\omega(n)$. MSB gives number of bits and LSB gives parity (eg: if number of prime factors is $124$ then since $124=(1111100)_2$ number of bits is $7$ and parity of $124$ is $0$). These cannot be $\#P$ complete or $\oplus P$ complete unless polynomial hierarchy has randomized subexponential algorithms and in $\mathsf{BQP}$.

These problems may be computable in randomized polynomial time since MSB and LSB on $0/1$ permanent is in $BPP$ and in $P$ respectively and $0/1$ permanent is $\#P$ complete.

- Number of bits in each factor if the number is a semiprime (there is no good algorithm to find degree of factors even for univariate polynomials).

Can there be efficient algorithms for these?

From 1. and 2. we can at best say if the number is a semiprime ($pq$) form or a triprime ($pqr$) form.

If a semiprime then 3. tells whether it is balanced (equal bits in $p$ and $q$).

We know that computing number of prime factors implies efficient factoring algorithm: The answers on the linked page make it clear that nothing like that is actually known. $\endgroup$ – Emil Jeřábek Sep 27 '17 at 13:37