How can a person's genotype for a trait be determined from his/her phenotype for the trait?

For round eyes you could not because like most physical features, there are a combination of genetic factors (pleiotropy) involved as well as environmental ones. Also genes that result in the same phenotype can have many permutations. What we do is more a process of elimination. We sequence and observe and occasionally experiment. Being dominant or recessive does not imply one is stronger than the other. It is more a mathematical abstraction than anything. Think of it as upper-case and lower-case to remove the connotations of the terms "dominant" and "recessive." That said, there are some phenotypes that are the result of a single base pair substitution and if that is the trait being observed than one can make conclusions about that genotype.