I see this statement all the time "half of our DNA (23 chromosomes) are from mom and 23 are from dad". Fine, but which chromosomes in our nucleus are from each parent? When I think of nucleus (before interphase), I imagine a total of 23 unique chromosomes and 23 copies of them. In the first cell after reproduction (zygote), does chromosome 1 from mom align with chromosome 1 from dad? When zygote divides (after S phase) do you now have 4 copies of the same cell? Since choromosome 1 from mom and dad is the same. What role does crossing over play in all of this. I have done lots of reading but nothing is helping me connect the dots.
First, to clarify on the number issue and parental origin of the chromosomes. For the moment, disregard crossing over. Then, you would begin with an adult somatic cell in the growth phase, having 46 chromosomes, two of each kind, that is, 23 pairs (assuming sex chromosomes to be of the same kind, including the case in males). Then, in the S phase, each chromosome replicates, creating exactly similar sister chromatids, each of these sisters being similar to the single chromosome the cell had in growth phase. Each of these chromosomes with two chromatids, sends one to each daughter cell, making the daughter cell exactly like their parents. In meiosis, after this step, one of each kind of chromosome separates into a gamete, so that instead of 23 pairs, each gamete has only 23 chromosomes, one of each kind. On fusion of two gametes, you would have the original configuration again.
Crossing over, just means that when the two chromosomes of a particular kind separate during meiosis, they exchange corresponding material. If no crossing over happened, the gamete would either have one of the two same-kind chromosome, or the other. But due to crossing over, you create a different set of genes on these two chromosomes, different from the ones fond elsewhere in your body. So if say two chromosomes of the same kind have genes $ABC$ and $DEF$, if no crossing over happened, the gamete would have either $ABC$ or $DEF$. But crossing over makes possible combinations like $AEF$, or $DEC$ etc.