And why LC instead of RC; is there an advantage of one over another?With an LC filter you do not get the drop in voltage that you get with an RC filter. This is because the resistance of the inductor is smaller than the resistance of the R you have to use.
Also an LC filter is a second order filter and the RC is a first order filter. The order of the filter is how quickly a low pass attenuates with increasing frequency.
I would have two separate ground plains and join them at one point.
There is no need for two supplies as shown here but it gives you the idea of two ground planes.