Are chromosomal abnormalities, such as trisomy, confined to certain parts of the body?

Usually. The vast majority of trisomy patients will have the trisomy in every cell of their body. There is a small percentage that will be what is called a mosaic, where the trisomy effects a fraction of the cells. In theory the trisomy starts in one group of cells and continues to all that develop after the change, but another normal line also produces normal tissues. The amount affects the outcome.
No. A down syndrome child can rarely be from normal egg & normal sperm. The fertilized egg divides into 2 normal cells, which themselves divide. If on one of the early cell divisions, one cell ends up with an extra #21 chromosome and survives to continue dividing, the child will be a mosaic with some trisomy 21 cells & some normal cells. The cells aren't in just some parts of the body, but everywhere.