What might determine whether a person who shows the dominant characteristic is homozygous dominant or heterozy?

If a genetic trait is truly dominant, there's no way to tell from an individual's appearance whether he/she is homozygous or heterozygous for that trait. Males tend to show sex-linked traits more frequently than females because we have one "Y" male chromosome we inherit from our father and one "X" female chromosome we inherit from our mother, while females have two "X" female chromosomes (one from Mom and one from Dad -- that he got from his Mom). Therefore, whatever gene we men inherit that's on our solitary "X" chromosome, that's the trait we will exhibit (even if it's rare and recessive), while a female who inherits some unusual recessive gene on one of her "X" chromosomes will almost certainly have it balanced out and overridden by the "normal" dominant version of that gene that's on her other "X" chromosome.