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Amending the soil with compost each year should be sufficient for fertilizing asparagus, or you may use a general purpose fertilizer.

Weeds should be controlled before asparagus sprouts. There may be some grass selective herbicides that can kill it in asparagus, but there are no organic solutions other than hand pulling.

It is usually better to cut them back in fall, but you can wait until very early spring if necessary. If you haven't read this already, the following article may be of some help to you for future reference: https://www.gardeningknowhow.com/edible/vegetables/asparagus/cutting-asparagus-foliage-back-in-autumn.htm

This indicates that they are under stress. Stress can come in many forms, from too little water to too little nutrients to pests on the plant. I would take a close look at the bed they are growing in and see if you can come up with some ideas on what may be causing their stress and working to correct it.

Feed the growing asparagus annually with a 10-10-10 granular fertilizer.

It sounds like you planted crowns, which means that there will be no germination. Germination would be when asparagus seeds sprout.

When you plant crowns, you are planting a dormant plant and you are waiting for them to break their dormancy. It can take 2-8 weeks for this to happen. Extra water and warmer weather will help this along faster.

You want to plant it at the same depth as what it was growing when you bought it. A deeper or wider container would be helpful for it, but in terms of how deep to plant it in that container, it should stay at the same depth you bought it at.