"Asymmetric molecules are necessarily polar"
Let's start by defining a polar molecule as a molecule with a dipole moment.


  Is a molecule with no symmetry necessarily polar?


Yes, a molecule with no symmetry belongs to point group $\ce{C_1}$ (e.g. the only symmetry element it contains is a $\ce{C_1}$ axis, rotation of the molecule 360/1 degrees converts the molecule back into itself).  The molecule is asymmetric and polar.


  Can a polar molecule still have some elements of symmetry (i.e. mirror
  image is the same as itself)? I think so ... because we can have
  achiral tetrahedral carbon molecules that are polar - i.e. methylene
  chloride. 


Yes, methylene chloride, vinyl chloride, methanol, chloroform, etc. all contain planes of symmetry, but are still polar.


  So, is it proper to say that a molecule that lacks symmetry is
  necessarily polar? 


If I understand correctly, this is a repeat of your first question.


  Or can a more nuanced statement based on symmetry be made? Is there a
  "minimum" level of symmetry needed to achieve non-polarity?


Yes


All asymmetric molecules are polar.  Such molecules belong to point group $\ce{C_1}$
Molecules belonging to point group $\ce{C_{n}}$ with n>1 are polar. 1,3-dimethylallene is an example (it has $\ce{C_2}$ symmetry, rotate it by 360/2 degrees about its $\ce{C_{2}}$ axis and it converts back to its original representation)
Molecules belonging to point group $\ce{C_{s}}$ are polar.  A sigma plane either contains or bisects molecules in this point group, and this plane is the only symmetry element.  Methanol, bromocyclopropane, vinyl chloride, etc. belong to this point group and are polar.
Molecules belonging to point group ${C_{nv}}$ are polar.  Formaldehyde and methylene chloride (both ${C_{2v}}$), chloroform (${C_{3v}}$) and hydrogen chloride (${C_{\infty v}}$) are examples.  This point group contains a ${C_{n}}$ axis and n sigma planes containing the axis.


If the molecule is not in one of these point groups, then it is not polar and does not have a dipole moment (or said differently, molecules in all other point groups [${D_n, S_n, C_{nh}, D_{nd}, D_{nh}, T_d, O_h}$ and higher] are not polar and do not have dipole moments).