Somebody on reddit did an excellent chart of the Falcon 9's landing accuracy:
While these landing positions are probably just approximate, I think it is reasonable to assume that the answer to the question is: yes, Falcon 9 and Heavy boosters need landing legs because they cannot land back onto their launch mounts.
Could you fit all six into a FH fairing? If not, would a slightly larger one be enough?
I can't find dimensions for GPS III sats, but this Lockheed Martin image shows how much of a Delta Medium 4-meter fairing is taken up by a single one:
Looking at the dimensions of the fairing in the Payload Guide, I think the satellite is about 5m long by 2.5m wide.