Is the 3-connected graph(s) on $n$ vertices with the minimum number of spanning trees always planar?
Edit. As it turned out I was not using the right switch for plantri.
This is therefore not an answer anymore but rather an extended comment for the case $n=11.$
As it turns out the minimal number of spanning trees of a 3-connected planar graph of order 11 is 3965 and is attained by the graph on the figure bellow.
As for the non-planar 3-connected graph I am yet to compute the answer. I'll post the result here as soon as it gets computed.