I'd say that the Italians would retreat into Tunisia and like Syria/Lebanon later rather than it impacting Vichy neutrality officially, it is agreed by all sides de facto that it will just be another front in the war, not that Vichy is in the war. It certainly does hurt German-Vichy relations though, like Syria-Lebanon did in 1941. Given British over extension to get to Tripoli, they were badly overextended to get to Benghazi IOTL, they'd get rolled very badly by Rommel when he showed up in Tunisia. Much like how he did them pretty badly in Cyrenica in 1941/42 he'd have an even better chance to do it in West Libya and cut them off completely. Rommel might even have a full corps of 3-4 divisions ITTL given the very serious nature of the situation politically, which would be extremely bad for British forces in Libya if that were the case, as they'd be in a pretty poor supply situation and Rommel a much better one thanks to the far greater logistics of Tunisia than most of Libya. Likely the German forces that would have been needed to conquer Greece are significantly lowered without the British there, while Crete is a cakewalk, while Rommel gets a major boost of forces and slaughtered the Brits in West Libya, but runs out of steam around Benghazi ITTL. He may or may not keep his enlarged corps, but that would certainly impact Barbarossa if he gets extra motorized/panzer divisions ITTL in 1941.
If the British have Tripoli, then Malta is almost certainly secure from invasion, because of the ease of resupply.
Germany and Italy would be capable of seizing Tunis, albeit at the cost of pushing most Vichy colonies into the Free French. Holding it would be tricky in the medium-long term, as British forces increased in strength and power projection, given the vulnerability of the supply line, but at least shipping to Tunis would be shielded by Sicily for most of the route.
Agree about Malta in terms of invasion, but X Fliegerkorps was nearly successful isolating it from resupply via aerial attack in 1941, but it was the Greek campaign that actually sucked off the Luftwaffe and loosened the aerial blockade enough to prevent it from surrendering. If ITTL the Brits don't aid Greece than the Luftwaffe anti-shipping air corps isn't needed in Greece, which means they keep up the blockade of Malta and it might well surrender ITTL due to lack of supplies. The Brits were considering letting it surrender until they got intelligence that the Luftwaffe had departed from Sicily, so sent in supply convoys. Had that not happened the Brits probably would have written Malta off in May 1941 and that is that for the island. The Brits holding Benghazi or Tripoli doesn't aid the ability to get convoys in given how attenuated supply was to forward elements of the British army in Libya as it was in 1941, who would need all the convoy assets the Brits can muster to keep them alive.
The Free French didn't exist in 1940 or 1941. That was mostly a consequence of the situation in 1942 after US entry. De Gaulle didn't even become the major French figure in exile until after Admiral Darlan was assassinated in 1942. Just like Syria-Lebanon didn't become Free French once the Germans and Italians moved in during the 1941 campaign to aid the Iraqi Revolt and the British invaded, the French fought the British tooth and nail in Syria. Even in 1942 they did in Madagascar too. So like the Syria situation IOTL Tunisia, which did work with the Axis IOTL in 1941, giving them trucks and fuel as well as promising to let them use their ports (the Axis never took them up on that), Tunisia doesn't trigger French colonial uprisings against the Germans; in fact given the antipathy toward the British in North Africa after the attack on the French fleet in 1940 in Algeria the North African French colonies would probably be happy to let the Germans and Italians kick the Brits in the pills so long as they don't have to be active combatants. Axis use of Tunisia, provided it doesn't result in Italian domination of the colony, would not be a problem at all, especially once the fighting moved on from the border. Except ITTL the Axis may continue to make use of the French ports and rail system in Tunisia to help supply Axis forces in Libya.