This is a really sophisticated question. I had to tap my friend the geneticist to help me out. I have edited this a bit.

Let's take the second question first. The short answer is "yes" for non-chromosomal DNAs and "no" for the nuclear genome. In general, small extranuclear genomes like mitochondrial DNA replicate more rapidly as they decrease in size. The more rapidly replicating a molecule, the higher its relative frequency in the DNA population, so over time selection favors deletions of genes that aren't essential and the extranuclear genome shrinks.

You can model this in bacteria with a strain carrying a large plasmid that encodes resistance to two antibiotics, and grow the cells in the presence of only one drug. After several generations, if you screen for the resistance to the second drug, you'll find that a proportion of the population is now sensitive and if you extract the plasmid DNA, you'll verify the presence of deletions. However, in higher eukaryotes unused genes in the nuclear genome tend to persist and accumulate in the course of evolution. In the human genome, most of the DNA is non-coding and a number of sequences are pseudogenes (code for a protein but lack the regulatory regions that allow them to be expressed. Look up "c-value paradox" if you're interested in this phenomenon).

In the case of mitochondria, the nuclear genome encodes mt-specific DNA and RNA polymerases (enzymes needed for replication), so with the evolution of those nuclear genes (or of the specialization of the enzymes they encode), replication genes on the organelle became non-essential. At this point, deleting the replication functions becomes advantageous due to smaller genome size. There is also a selective advantage for the host to control mt replication - it can vary the number of mitochondria so that it only invests in as many as it needs. The combination of these pressures would be a relatively strong selection for loss of self replication.

You may be interested in studying genetics or molecular biology.