Any periodic signal can be seen as a sum of sinusoids (Fourier). We have a fundamental frequency plus an number ideally infinite of harmonics weighted in amplitude. A low pass filter will eliminates (o reduce in amplitude) a given number of harmonics depending on how the cut-off frequency of the filter is far from the fundamental. If the reduced hamonics are far from the fundamental the signal shape will be close to the original. Moving back the frequency of the filter towards the fundamental, the edge of the square wave will became more and more smoothed. Finally, if the frequency of the filter is close to that of the square wave, we will obtain a sinewave. However you have to define where, in frequency, and how much in amplitude the filter has to operate. Furthermore you have to define what's the acceptable smoothing of the edges. Also the order of the filter has an effect.
About the ringing this is due to the high Q; you can reduce (or eliminate) it f.i. adding resistors in parallel with the inductors. It has a cost, of course, mainly in terms of filtering effectivness.