maybe separate clashes instead of one all out war? One against Syria over the water, and another with Jordan later/sooner over the raiding?
That's definitely possible. As it was, Jordan almost didn't involve itself (Israel would almost certainly have respected its neutrality).
The problem is, as I said in my first post, in 1967 Israel's running low on spare parts. Some people estimate they had only a few months' flight left, and it's uncertain whether they would have been able to fly two big campaigns, so there's a strong incentive to attack both Egypt and Syria at the same time, if there's any question that the other might attack if Israel attacked the one. Now, considering the presence of the UN in the Sinai before Nasser politely asked them to leave in mid May, it's possible that Israel would have just attacked Syria, believe that Egypt would stand by. For Egypt to stand by, though, would have been the final nail in the coffin of Egyptian claims of leading the Arab world. As Nasser was currently in the process of being embarrassed in Yemen, it seems unlikely that he'd want to let that happen...or, at least, it's likely that fairly paranoid Israeli generals would have felt that way.
In theory, I guess, with a UN peacekeeping force that doesn't completely fail at their job, the 6 Day War could have ended up just being an engagement with Syria, likely still ending in the annexation of something like the OTL Golan Heights. It's possible that without other distractions, Israel could have pressed a little further, but honestly there's not much to press - the present border is about 50km from Damascus, as the crow flies. And the Golan was fairly thinly populated, and with a lot of Druze; every further kilometer is more and more "regular" Sunni Arabs. Plus the Golan is a great defensive position - they form a pretty distinctive little plateau.
I still see conflict with Egypt in the future over Fedayeen raids from Gaza, though...though I guess Egypt could at some point actually cut them loose, in which case Israel would probably occupy them pretty quickly unless they could get a UN occupying force.
Funnily enough, if Egypt waits until Israel blows its load over Syria, and then invades even a few weeks later, they might catch Israel completely flat footed and might even be able to roll over Tel Aviv (though I suspect that Israel would be able to stop them around Netanya, especially if Jordan didn't cooperate with them - still, that's basically 2/3 of the Israeli population and like 80% of their land). I can't see Egypt withdrawing short of a UN sponsored intervention.
And Jordan remains tricky, though I guess that the conflict there could be kept to a low simmer until something like Black September happens - which, since this Jordan will have way more Palestinians, and more territory for the same smallish army to cover, probably go to the Palestinians, resulting in the establishment of the Republic of East Palestine where the Hashemite Kingdom of Jordan was. Honestly, Israel did Jordan a favor by conquering the West Bank (and Jordan knew that, which is why they refused its return).
Regarding the French quite a bit resides with De Gaulle's perceptions... and given the man it is questionable that anyone sans himself knew what exactly he had in mind. Was it just rational strategic calculation against French common sentiments? Was getting ticked of by Israeli actions? Something in between? In all cases beyond "deliberate plan to dump Israel" which appears somewhat unlikely, survival of the Franco-Israeli relationship is probably within the realms of plausibility.
Everything I've read says that De Gaulle actually really liked Israel and the Israelis. The issue was more one of realpolitik. In 1967, France had relatively recently pulled out of most of its African colonies, many of whom were Arab. It wanted to capitalize on its former colonies as potential economic partners - after all, they already spoke French and were used to French standards and such, and their militaries used French equipment, etc etc. But a lot of the Arab countries weren't so happy about France being Israel's main patron, so France had to choose between Israel and basically the whole Arab world.
As far as I can tell, "it wasn't personal, it was just business".
If France doesn't pivot towards the Arabs, there's no compelling reason that they and Israel can't stay friendly for as long as you want.