What's your question? Ask

Do you have a picture of the plant? We would need to identify it before we can give you advice on how to propagate it.

No, those parts of the roots have not vegetative capacity. The best thing to do with them is add them to the compost pile.

For rosebushes, you want to take cuttings after the rosebush has bloomed and take cuttings from a cane that has just bloomed, as those canes are considered mature canes. Here is a link to an article about starting rosebushes from cuttings: https://www.gardeningknowhow.com/ornamental/flowers/roses/roses-from-cuttings.htm

A florescent light bulb set just a few inches from the cuttings will help to provide supplemental light. Here is more info: https://www.gardeningknowhow.com/houseplants/hpgen/fluorescent-lighting-for-indoor-gardening.htm

As for potting media, this article will help: https://www.gardeningknowhow.com/garden-how-to/soil-fertilizers/soilless-growing-mediums.htm

This generally depends on the type of plant and cutting you have. Cuttings from herbaceous plants can be taken anytime. Softwood cuttings are best taken in late spring to early summer, while hardwood cuttings should be taken when plants are dormant during fall and winter.

The answer to your question is 'yes'. A plant that is propagated from another via a cutting could outlive the plant that it was originally propagated from.

For more information on plant propagation, please visit the following link:
https://www.gardeningknowhow.com/garden-how-to/propagation/propgen/types-of-plant-propagation.htm