X-Men: First Class has the coin used by Sebastian Shaw which later is used by Magneto to kill him, himself succeeding the villain he fought against. Cerebro doesnt appear, because it was constructed as a prototype in the film.
Dark Phoenix, while being the most toned-down title sequence (no Cerebro or mechanism), features fire in the film title.
The connection between The Gifted and Legion and the X-Men films can be found if you take into account a recent episode of Legion where David talks about the theory that their are strings of alternate timelines that diveverge and diverge at different points due to different decisions.
Past Xavier (McAvoy) seems to be a lot more cocky and overconfident since Days of Future Past. Well, obviously: hes changed history, defeated a mutant god and mentally controlled the mutant who defeated the aforementioned mutant. With that said, his overconfidence does get the better of him on those occasions.
Future Xavier (Stewart) never went through those events. So he appears more sober and serious, and doesnt underrate anything or anyone. His battle with the Phoenix in X-Men: The Last Stand was for the first time and he came off the worst for it, because he never had previous experience with it or anything like it.