I've never been to Paris, and only five days to another place in France, but I think the thing about eyecontact could be similar to what we (sometimes) do in Sweden:
When you approach a stranger (opposing directions on the same path sort of) you look them in the eyes for a short while (less than 1 sec), then look away, then give a very short glance just before you pass them. Since I've done this since childhood I don't know why, but my interpretation is that it's a way to acknowledge that the other person is there/existing (and a way to say "I see you and we pass and none of us are dangerous"). If you are alone in the forest and meet it's rather common to say hello. On the other hand when walking in a crowded street this doesn't work so people don't look into other people's eyes very often there.
Would be interesting to hear a French explanation, if it's something similar?