Alfalfa is a good cover crop, but like sun hemp and pigeon pea, it has relatively deep roots and not the easiest thing to remove if it is kept too long. It is hard to dig up by hand.
I prefer to use other legumes like fava beans, cowpeas for nitrogen fixing and buckwheat for biomass. Remember a true cover crop only benefits the soil if it is able to develop the nitrogen fixing nodules on the roots and if the nitrogen stored in the plants are returned to the soil and not lost to harvesting. Cover crops need to be turned in when they start to flower for maximum nitrogen return. You also need to turn over most cover crops because they are by their very nature weedy. For me, the soil in one of my plots is extremely alkaline and fungal dominant so legumes need to be inoculated first to get any kind of nodulation. Normally, the rate without inoculation is 0-4 nodules. Even double cropping after inoculation, the level falls back to baseline. With inoculation the average is 8-12 nodules.
If you live in an area with snow, and your cover crop is not winter hardy, then the problem is solved. Nature will do the killing and over the winter the nutrients will sequester and slowly release as it decomposes.
https://www.growveg.com/guides/green-ma ... er-garden/
https://www.growveg.com/guides/green-ma ... -the-ugly/
Luckily, I live on land that was once a pineapple field so rocks are pretty rare, and the land is mostly fill. I do however, have a red clay oxisol.
The depth can be anything you like 18 inches to 2 ft will get most of the root zone for the majority of your root crops unless you are planting parsnips or gobo. If you have clay, or your garden is being located in a low spot, you should check the drainage by doing a percolation test. Rocks in the soil will actually help drainage, you just don't want too many of them where your plants will be growing.
https://www.dummies.com/home-garden/gar ... -drainage/
Improving clay soil can usually be done by adding organic matter. Organic matter breaks down and will have to be added regularly. Over time the organic matter will fluff up the soil, improving soil tilth, and depending on what kinds of compost and manures you add, you can effectively change or stabilize pH over time. Chicken manure will alkalize acidic soils, and Peat moss will lower the pH of alkaline soils.
Usually I mix equal volumes of compost with soil. Unless I am going for a specific effect, like acidifying the soil, I usually use a blend of different composts and composted steer manure. I test my soil about every 3 years. Compost does contain phosphorus which I don't need, but all the fertilizer I actually add is sulfate of ammonia. I do use synthetic fertilizers, I don't like to use animal byproducts, even manure in quantity. Manures contain salts as well and more than just the nitrogen I need.
https://garden.org/learn/articles/view/1310/