Common modulus attack with not coprime exponents
No, it's not possible (or so we hope).  If you could, you could break RSA.

Suppose you had an Oracle that, given $n, e_1, e_2, m^{e_1}, m^{e_2}$ with $\gcd(e_1, e_2) = d$, and which is able to output $m$.  We can assume that the Oracle only works for a specific $e_1, e_2$ pair.

Then, suppose you were given $c = m^d$, and wanted to recover $m$.  Here is what you could do:


Compute $c_1 = c^{e_1/d}$, and $c_2 = c^{e_2/d}$


We note that $c_1 = m^{e_1}$ and $c_2 = m^{e_2}$.


We give $n, e_1, e_2, c_1, c_2$ to our Oracle; our inputs matches the Oracle requirements and so it produces $m$, thus solving the original RSA problem.