During the cell cycle the cell replicates almost everything inside itself, organelles, the MTOC, and it's DNA. These chromosomes then condenses and pair together to form chromatid. These then line up on the metaphase plate (cell middle) and are pulled apart by the mitotic spindle. They are in theory exact replicas of each other (if not this causes problems like cancer). Each daughter cell should receive a full copy of the genome, with one of each chromosome. If it doesn't due to a fault in the division this can lead to aneploidy which is the wrong number of chromosomes in a cell which is caught causes cell death, if not can lead to cancers.
With aging I think you are getting confused with the telomeres. These are at the ends of the chromosomes and consist of repeats of DNA code (usually G), the protect the coding part of the DNA from damage by acting as a sacrificial buffer region. It may also form and interesting non Watson-Crick double helix structure. As the cells divide sometimes not all of this is replicated, or is damaged causing telomere shortening. You have and enzyme telomerase which is responsible for correcting this and maintaining this buffer zone of DNA.
Telomere - Wikipedia, the free encyclopedia
If you have an academic log in look at this
Access : : Nature
In some immortal cancer cell lines telomerase is unregulated and it is thought this might be one of the reasons for the cell lines immortality. It is a hypothesis that telomerase activity and telomere length is somewhat related to aging.
You are right about the DNA it is made up of coding regions for proteins (cell machines), the genes and this is interspersed with non coding or 'junk' DNA. mostly made of tandem repeats.
However this view has been counter recently with some research pointing the 'junk' DNA does serve a purpose. It is more correctly known as satellite DNA. Some of it is promoter regions (gene traffic lights) and some areas encode for 'dead' proteins that no longer get transcribed (turned into proteins).
Tandemly Repeated (Satellite) DNA
I hope this helps, if you have any questions or want more detail just post back :)