The thing is, to accomplish his overall goal, things didn't have to go exactly to plan.>He wanted to tear up the Avengers from the insude, but did he even plan the Sokovia Accords or everything just worked casually on his favor?
After the debacle that was Ultron and Sokovia it was only a matter of time until the world governments would try and regulate the Avengers in some way. Zemo used Hydra and Crossbones as the trigger for that, but sooner or later something would have done it.>How the fuck did he know Bucky killed Tony's parents?
It was in the leaked Hydra documents Black Widow leaked at the end of TWS. Early in the movie Zemo says he painstakingly decrypted and reviewed all of them.>And how the fuck could he ever know that only Tony, Steve and Bucky would conveniently be the only ones to follow him to the snow base? What if only Steve and Bucky were the only ones to arrive there? He couldn't have shown the tape to Tony.
He called the hotel so that the attendant would find the dead psychologist, thus making it public that Bucky wasn't the one who bombed the UN conference in Vienna. From there it's reasonable to expect that AT LEAST Cap, Bucky, and Tony would come, if not more Avengers. If Tony doesn't show Zemo could always retreat and show him the footage at another time.
If it seems unlikely that he could predict all of these other people actions, remember that he's been doing nothing but working on this and listening to his dead wife's voice mail for 2 years. That includes studying each member of the Avengers for weaknesses and motivations, which he could do easily with the leaked information and the modern internet-powered news cycle. He was obsessed with Cap in particular, to the point he noticed a slight difference in eye color from what he expected when they met in person.