The exchange of genetic material happens during the prophase I of meiosis. We know that since meiosis I involves the pairing, crossing over, and separation of homologous chromosomes, our answer must be in meiosis I. During metaphase I, the tetrads line up along the metaphase plate, and during anaphase they begin to be pulled to opposite sides of the cell by spindle fibers.

In prophase I, homologous chromosomes pair up and facilitate the exchange of genetic information through the process of crossing over.

In preparation for cell division, DNA is copied (replicated) to create chromosomes with two identical sides (sister chromatids), which are connected at the centromere. DNA must be replicated before prophase, in S phase of interphase. Do not be confused with the terms centrioles, centrosomes, and centromere. Centrioles are located at centrosomes, which are found at opposite poles of the cell during meiosis stages of prophase, metaphase, anaphase, and telophase. The question is asking about interphase, which is prior to the meiosis/mitosis stages. 

For homogenous or identical cells of a particular tissue, mitosis allows for the restoration of lost or damaged tissue by resulting in identical daughter cells which can replace lost cells. The same is true for an asexual population. The process of mitosis has several checkpoints to ensure fidelity with each round of copied cells. Meiosis, on the other hand, tries to ensure evolutionary fitness of a sexually reproducing population by introducing a variety of genetic alleles per gamete which will result in fertilization to create a unique individual. Therefore, mitosis allows for growth, restoration of damaged tissue, asexual reproduction (for example, bacteria!) and does not result in genetic variety.

Meiosis, which includes independent assortment of homologous chromosomes and chromosomal crossover, contributes to genetic diversity. Independent assortment of chromosomes is the random distribution of one chromosome per homologous chromosomal pair to each daughter cell during anaphase I. Chromosomal crossover, which begins in prophase I, results in genetic recombination between each tetrad (homologous chromosomes). Nondisjunction is not a correct option as this typically results in large scale genetic abnormalities and infertility which subsequently does not contribute to a population's long-term genetic diversity. 

Human female gametogenesis occurs through meiosis and results in three non-functional daughter cells known as polar bodies and one functional and much larger daughter cell called an ovum. Unequal division of cytoplasm during meiosis I and meiosis II towards only one of the daughter cells results in the ovum. Non-disjunction is the unequal assortment of chromosomes, not cytoplasm. Human female gametogenesis results in ovum while human male gametogenesis results in sperm. Mitosis does not result in gamete daughter cells but identical daughter cells as mitosis occurs in somatic (body or non-germ cells) cells. 

Germ cells give rise to gametes through meiosis. All of the other organisms listed are asexually reproducing organisms and therefore only undergo mitosis. 

Nondisjunction of homologous chromosomes may occur in anaphase I and nondisjunction of sister chromatids may occur in anaphase II. 

Independent assortment depends upon random alignment of homologous chromosomes in metaphase I which will result in randomly assorted chromosomes into two daughter cells at the end of meiosis I. Crossover, or recombination, occurs in prophase I which contributes to genetic diversity, but does not affect random alignment of homologous chromosomes. Spindles align chromosomes randomly in metaphase I, not anaphase I during which they are already "aligned" and are moving towards opposite poles of the cell.

In meiosis, the gametes are haploid having half the number of chromosomes of a diploid cell.  The gametes are the egg or sperm cells that combine in sexual reproduction.