The LC second-order (low-pass) filter is effective at shaping a waveform into a sine wave, if resonant frequency matches incoming frequency.
L & C values need to be customized to your load and frequency.
It's possible to build a sine-wave oscillator which automatically detects LC resonant frequency, and creates oscillations at that frequency through your load.
Of course there's a challenge operating at 5 GHz more so than at slower frequencies.
I think the output of your VCO is not a pulse, but it is a square-wave signal.
Yes, you can use an LC filter (LPF or BPF) at the output of the VCO to make it sine wave.