All Birds are dinosaurs, but not all dinosaurs are birds. All dinosaurs are reptiles, but not all reptiles are dinosaurs. Crocodylians for example, are not dinosaurs

I'm not sure about chickens specifically, but as you can see above, all birds are part of the theropod clade, which is the same clade as T-Rex, so they are indeed closely related.

Convergent evolution tends to happen when animals face similar environmental challenges, because sometimes a certain body plan is simply the most efficient way to tackle these challenges. For example, if animal A and animal B both live in the water, and they both need to swim fast to fill their niche, they're likely going to evolve a similar streamlined body with dorsal and pectoral fins. Ex. Sharks, Dolphins and Ichthyosaurs.

My favorite example of how cladistics has changed our perception of animals is the elephant shrew.

We named it shrew because it's obviously a shrew, right? And the elephant part was just a funny little joke because of his big snout, even though it obviously isn't related to elephants! Well, it turns out that not only is it NOT a shrew, it is actually one of the closest living relatives to elephants! How crazy is that?