Like all vegetables, asparagus thrives in fertile soil, so an annual mulch of compost/manure suits it well. Especially if the soil is light because I have noticed how it thrives better on clay than when I grew it on brashy soil. Organic matter helps to hold moisture, and planting on flat ground rather than ridges helps too, as well as making weeding easier. As long as you are planting into soil that is clear of perennial weeds, and after a year of hoeing/weeding to clear residual weed seeds, there should be little weeding to do in subsequent years. But do it, to prevent any weeds seeding.
Regarding variety to grow, I got in a muddle with mine here and am not sure which is which, but they have all done well. Any that claims to be productive is good because one seldom has enough!