Not only are they not in the same genus (Gallus is a genus, not a phylum, for the record), they aren't in the same family (tinamous being in Tinamidae and chickens in Phasiamidae) nor in the same order (tinamous being in Tinamiformes and chickens in Galliformes), nor yet in the same infraclass (tinamous in Palaeognathae and chickens in Neognathae). Their nearest related grouping is that they're both in the same class, Aves. Basically, they're both birds and that's the only way in which they're related. To put it into context, ducks are more closely related to chickens than tinamous are, and it is well known that ducks and chickens cannot interbreed. I would highly doubt that breeding a tinamou and a chicken together would even produce a fertilized egg let alone an embryo or beyond.
As for care and housing for tinamous, I wouldn't even be able to guess on that. I was unaware they were kept anywhere but in zoos outside of their natural range.