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the best time of year to plant a asparagus bed would be early to mid spring.
asparagus plants like cool, moist conditions. make sure that when u plant your asparagus, give the plants plenty of water and also mulch around each plant, about a centimeter to 1 inch thick.

happy planting!

Here is information on planting and growing asparagus:
https://www.gardeningknowhow.com/edible/vegetables/asparagus/planting-asparagus-how-to-make-an-asparagus-bed.htm
https://www.gardeningknowhow.com/edible/vegetables/asparagus/cutting-asparagus-foliage-back-in-autumn.htm

No, they do not do well in the shade. They need 6 or more hours of light to do well.

These articles will help you:
https://www.gardeningknowhow.com/edible/vegetables/asparagus/planting-asparagus-how-to-make-an-asparagus-bed.htm
https://www.gardeningknowhow.com/edible/vegetables/asparagus/cutting-asparagus-foliage-back-in-autumn.htm
https://www.gardeningknowhow.com/edible/vegetables/asparagus/transplanting-asparagus-plants-tips-for-how-to-transplant-asparagus.htm

You can apply fertilizer still. It will be fine.

You can move them, but you will, of course, not be able to take a crop from them this year. Dig up a clump, making sure that you dig a good foot or more down, if possible. Wrap the rootball in a damp towel and then wrap that in a trash bag. This article will have other tips for transplanting it:
https://www.gardeningknowhow.com/edible/vegetables/asparagus/transplanting-asparagus-plants-tips-for-how-to-transplant-asparagus.htm

Once you get to your new location, plant it as soon as you can. Make sure it stays well watered and watch for transplant shock. This article will help you keep an eye out for that:
https://www.gardeningknowhow.com/plant-problems/environmental/learn-how-to-avoid-and-repair-transplant-shock-in-plants.htm

Good luck!