The first two expressions are correct and usually denote components of a 2nd-rank tensor. You can lower and raise indices with the metric components ##g_{\mu \nu}## and ##g^{\mu \nu}##, respectively, i.e., you have
$${A_{\nu}}^{\mu} = g_{\nu \sigma} g^{\mu \rho} {A^\sigma}_{\rho}.$$
The last expression should be avoided, because the horizontal placement of the indices is not indicated. It's ok if the tensor ##A## is symmetric, i.e., if ##A_{\mu \nu}=A_{\nu \mu}##, because (only!) then
$${A^{\mu}}_{\nu} = g^{\mu \rho} A_{\rho \nu} = g^{\mu \rho} A_{\nu \rho} ={A_{\nu}}^{\mu},$$
and the horizontal ordering is not important.