How come "Our cat do not eat sugar" is incorrect? When on the previous lesson it said asked to type "Meus cavalos nao comem arroz" and I typed in "My horses does not eat rice" and it was incorrect? Why is it different?
In both cases there is a mismatch between subject (singular/plural) and the form of "do" used. You say "cat does" (not "cat do") and "horses do" (not "horses does").
When is it appropriate to use 'Nosso' as opposed to 'Nos'? Are they interchangeable?
Not really. "Nós" means "we" and "nosso" means "our" (when talking about a masculine singular thing). There is another word "nos" (without the accent) which can mean "us" among other things.