The following website gives information on cover crops to use in gardens.  Annual clover is a good one for fixing nitrogen in the soil. And it will be gone in spring when you are ready to plant again.

https://extension.umn.edu/how-manage-soil-and-nutrients-home-gardens/cover-crops-and-green-manures

Yes some will. But as the article states, you kill it when you till it back into the soil.  If you want a cover that you don’t have to till and will die over the winter use rye grass or oats. However, they grow best in cool weather so they may not work as well if you are harvesting in hot weather and replanting cover immediately.

Here is an example of an annual clover that grows well in sandy soil. Although it usually is killed by low temps, if any survives you will need to till it. You are most likely to find it at a farm store or google “crimson clover” and click on the shopping tab.
https://www.canr.msu.edu/news/management_and_planting_of_crimson_clover

Yes, just be aware that you will have extra nitrogen in the soil.  So you do not want to add too much nitrogen fertilizer.  If you add too much nitrogen the green beans will grow greenery at the expense of beans.