While it seems you've gotten it to work, I'd suggest forgoing the whole custom SlashNode class thing. All you really need to do is this:
EntranceWebNode c1_s = new EntranceWebNode(cave1_start, "Cave entrance", "Enter");
cave1_start is the Tile, "Cave entrance" is the entityName, "Enter" is the action. To adapt this to your case:
private static final Tile web_tile = new Tile(3210, 9898, 0);
EntranceWebNode web = new EntranceWebNode(web_tile, "Web", "Slash");
You can then make the connections.
I'm also not a fan of how you're hard coding the connection between your custom path to the global nodes. Here's an example of how I've done it:
p6.getEnd().addDualConnections(WebFinder.getWebFinder().getNearestGlobal(p6.getEnd().getTile(), 10));
Since you only have one node, you can specify the tile in getNearestGlobal to be a tile close to the global node you want. This way, if the nodes numbering ever changes, you won't have to go in and change it manually. To adapt this to your case (these are the real tiles I grabbed from Expliv's Map):
private static final Tile north_tile = new Tile(3210, 9903, 0);
private static final Tile south_tile = new Tile(3210, 9894, 0);
web.addDualConnections(WebFinder.getWebFinder().getNearestGlobal(north_tile, 10));
web.addDualConnections(WebFinder.getWebFinder().getNearestGlobal(south_tile, 10));
Finally, add the node:
WebFinder.getWebFinder().addWebNode(web);
Last comments: You can add the web nodes in onStart(). I'm not sure why you'd want to add them at the leaf instead of onStart().