nice curves!
converting k points to k value:
The k points are relative to 2pi/a, so you can simply multiply 2pi/a to get k value.
modes:
you need to use frequency domain monitor to record 1 or 2 frequency points on the band, and then check the mode symmetry property. Please note that now you will need much longer simulation time, and most likely need to use apodization: https://optics.ansys.com/hc/en-us/articles/360034902473-Understanding-time-apodization-in-frequency-domain-monitors
Since you only need to know the symmetry property, simulation time may not be too long with apodization. You can test how long it is proper.
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