After a plant is removed/dies, how long does it take for the roots to break down?
Really, depends on conditions.
They can last years (dead) in a free draining mix of peat in a pot, but can rot in weeks in a healthy, damp, garden loam. Usually, the healthier the soil, the faster it will break down roots. Non soil growing mediums usually take much longer. Other factors in play include temperature (the warmer the faster), moisture (soggy soils and very dry soils slow microbial action), levels of soil microbes (highest in well manured, well cared for land in long use, and in virgin soils. Also, some plants' roots live underground for a good while after the top is removed, before they die (assuming they don't regrow).
Do they eventually provide viable nutrients?
Not in any quantity that you will notice any difference in the plant growth. It will only put out what nutrients are in the roots, so not many at all, for most small plants. The organic matter added will be the most beneficial addition. With one exception. Legumes (beans, etc.) will provide a nice nitrogen kick that other plants won't.