Fallout 4 - Materials not being shared

You need to work both ways sometimes theres a glitch where some items appear to link due to modding at this piont i have ten penny one way to sactuary and i cant share workshop items hope that helps

What can I do to make sure settlements are actually connected?

Caravans do get occasionally stuck and indeed there are glitches that break them down, it's not a very stable mechanic and I have no definite answer what's causing it. I have some vague memory about power losses in one settlement causing problems, because beacons etc go offline. Or maybe it is missing something else like food or water, so that the caravanner can't work. If a settlement is attacked and you don't respond, raiders will break things up. Including, power, food or water. You'll have to repair the resource if that happens.

You can also try to hunt down the person assigned as caravanner and re-assign them the same route. If they get involved in fights with raiders etc that might also cause problems, but that only happens when the player and the caravanner are both near hostiles.

Also avoiding excessively long routes seems wise, I never had much problems when I just connected all settlements to the closest one and built up a network like that, rather than having most of them go to lets say Sanctuary.

Is the Island actually capable of being connected to the Commonwealth through supply lines?

Yes you can connect to Longfellow's cabin in Far Harbor and also to the gas station on the Nuka World DLC map. It works like any other trade route, though I always picked a settlement near the DLC map. For Far Harbor, the nearest would be "Coastal Cottage" and for Nuka World, "Sunshine Tidings". Not sure if it matters.