Hand cream is designed to moisturise the skin of the whole hand, whereas cuticle oil focuses on the skin around the nail and on the nail plate itself, which need specific nourishment and flexibility. They complement each other well: apply the oil to each cuticle first, then the cream.
Pushing them back is the most advisable option. The cuticle protects the base of the nail and, when you cut it, some of that protection is lost and the skin tends to grow back thicker and rougher. A cuticle remover softens dry tissue so you can gently push it back and remove it, with no need for nippers.
It is usually due to dryness: frequent contact with water, soap and cleaning products, the cold or continued use of nail polish remover reduce the nail's flexibility, until it ends up splitting into layers. Moisturising the nail and cuticle every day and using a strengthening treatment helps it hold up better.
The fuller routine, with cuticle remover, filing and treatment, only needs doing once or twice a week. Moisturising, on the other hand, is daily: a few drops of cuticle oil or gel in the morning or evening keep the nail flexible between manicures.
Cuticle oils and gels can be applied around the nail even when it is painted, as long as the polish is completely dry. Treatments vary: some are used as a base under colour and others are designed to be worn on their own, so it is best to check the directions for each product.