Island sink and DW venting

I am looking to purchase a home and have run across a couple of plumbing issues. I don't believe the kitchen sink is vented properly or vented at all. Assuming the sink is vented, the distance from the p-trap to the where the vent ties in is 10 to 12 feet at least. Is this okay is there a chance that trap could be siphoned out?

An AAV would solve the issue pretty easily, but from what I have read they are a last resort. And I am not sure if they are even allowed. The home location is in Northport, NY.

I understand that the island should be plumbed like the image below and that makes sense. In order to not have the plumbing hanging to low in the basement and for the drain to be high enough to make it to the main stack without have move where it enters the stack how close (vertical distance) can the vent and drain be where the red arrows are?

You are in NY. I believe you can go with an AAV and though I hate the damn things, an island sink is the only place I would rather see one than a loop vent. See that horizontal section on the loop vent that leads up? That flat section is going to get filled with crap and block the vent and although you are supposed to put a clean out there, nobody is ever going to periodically use it. The other solution is to waste stack vent it if possible.

Good point on the eventual clogging of the horizontal vent section. I did not think of that. In this case the horizontal section would be a decent length (10 to 15 feet).

Just curious. Would up-sizing to 2 or 3 inch (depending on what's downstream) help on the horizontal vent and by how much?

For that horizontal on the vent to plug up, you would first have to plug the entire kitchen sink line. If the main kitchen line is plugged, it's going to get snaked first.

The main lines plugs, and then nothing is done. Really? Doesn't someone figure out that the kitchen sink has quit working at some point? I've seen plenty of Island Sink vents put in over the years. I like em!

The only way ANYTHING gets into the horizontal vent is when the sink plugs up. When the drain line is cleared, the water in the vent DRAINS carrying the 'debris' with it. For all practical purposes the vent line should NEVER plug up. A smaller vent line will work better than a large one because it will "scour" better when the drain is cleared.