Some things.
1) The Books were closed in 1957 for NYC, but the families were still making members in 1958, this was due to guys recently being released and the other families finally gave the ok for prospects. Even when the books were closed the bosses did make a few guys during that time. This was mainly New York Families, but other families followed suit in closing their books as a sign of solidarity especially after Appalachian. The only families that actually continued to make members during the books being closed were smaller families, exception being Philadelphia and Decavalcante who closed their books sometime in the mid 1960s, and the Commission told Bufalino, Marcello, and Lanza that their books could remain open with no interference from the other families. San Francisco and New Orleans out of respect for how old they could trace their roots, while Russell Bufalino was just respected all around, we know that the Colombos sent guys to Pittston and Scranton to be made and transferred back, Jerry Catena sent someone over there to be made, and Russell made a couple of guys for Tommy Lucchese.

2) Giuseppe Buccellato was in the D'Angelo crew and where his capo went, he followed. Not a lackey, but had strong connections to Sicily, D'Angelo went with Digregorio and his soldiers including Buccellato.

3) Gambino actually mad Bonanno look bad to Commission members. Wiretaps from The Genovese, Rhode Island, and Decavalcante paint a good pitcher of it as Bonanno didn't show up for Commission meetings, but wiretaps and informants have shown that Bonanno was not informed or was mislead by Gambinos messages.

4) The Bonannos today still have those connections to Trapani province but the connections you talk about are gone mostly as they do not hold as much power as they once did, but the Bonannos are in contact with those in charge. Bonanno and Gambinos are the only two New York City families with strong Sicilian ties.