We've already had an answer from the point of view of crystal field theory. This is actually more or less equivalent to your solution that six electrons are taken by ionic means (which is an oversimplification of the meaning of oxidation number +6, but not a bad place to start thinking about this) leaving one unpaired electron to explain the paramagnetism.
At the other extreme of oversimplification, you could model this using a simple valence-bond model based on the Lewis structure (see Wikipedia), which I think is what you may have in mind in your second question. In this model, since there are formally two single and two double bonds, you would say that there are four sigma bonds formed by Mn $4s3d^3$ hybrid orbitals, and two pi bonds formed by the remaining two Mn $3d$ orbitals. These six orbitals together contain seven electrons from the Mn and four from the four O atoms, for a total of eleven electrons, again leaving one unpaired.
In practice, for what it's worth most research chemists would use neither of these pictures, but instead a molecular orbital model.