Michael had been horning in on Roth's gambling empire since he returned from Sicily. Roth tricked Michael by pretending that Michael was his heir-apparent--all the while buying time for an attempt on his life. Anthony's party was the perfect opportunity. Roth knew Frankie Pentangeli would be there, contentious over Michael's support of the Rosato brothers against him. So Frankie would have the perfect motivation to kill Michael, and thus would make the perfect fall-guy for the crime.
But the scheme was just a little too pat for a guy with Michael's Sicilian cunning. Michael figured out, right after the shooting, that Frankie was too small-time, too lacking in ambition and imagination, to have gained anything by killing him, considering the risks. But Roth would have the most to gain: eliminating his competitor in the gaming business, putting his allies, the Rosatos, in charge of Michael's NYC operation--and having Pentangeli take the blame. It was perfect for Roth--and Michael saw through it.

Re. Vito and Barzini: In addition to what Don Cardi said, I'd add that Barzini gives himself away by sitting at the head of the table at a meeting that Vito called and nominally chaired. He also (nice touch!) was the only guy smoking a cigarette (the others were puffing on DiNobli's)--a giveaway that he was the modern guy. And, as Vito said, "Tattaglia's a pimp; alone he could never have outfought Santino." The only surprise here is that Vito hadn't figured it out earlier.

Others have said that Tony Sirico appeared in the Trilogy, but I don't recall where.