(a) The term homogametic and heterogametic refer to the organism depending upon whether all the gametes contain one type of sex chromosome (Homo = same) or two different types of sex chromosomes (Hetero = different). Humans show XX/XY type of sex determination, i.e. Females contain two copies of X chromosome and males contain one X and one Y chromosome. Therefore, ova produced by females contain the same sex chromosome, i.e. X. On the other hand the sperms contain two different types of chromosomes,
i. e. 50% sperms have X and 50% have Y chromosome open from half the autosomes (Meiosis). Therefore, the sperms are different with respect to the composition of sex chromosome. In case of humans, females are considered to be homogametic while males are heterogametic.
Yes, there are examples where males are homogametic and females are heterogametic. In some birds the mode of sex determination is denoted by ZZ (males) and ZW (females).
(b) As a rule the heterogametic organism determines the sex of the unborn child. In case of humans, since males are heterogametic it is the father and not the mother who decides the sex of the child. In some animals like crocodiles, lower temperature favour hatching of female offsprings and higher temperatures lead to hatching of male off springs.