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Coreposis and Fire Spinner ice plant are both winter-hardy in your zone, so the cold alone wouldn't have killed them, but there could be another problem with the site. Ice plants prefer dry sites, since they are succulents. An overly wet or shady spot, or especially a wet winter, could kill them. Here is more about ice plant care:
https://www.gardeningknowhow.com/ornamental/flowers/ice-plant/growing-hardy-ice-plant.htm

Full sun and well-drained soil are also best for coreopsis, but it is a more durable plant, so it may still show up later in spring.

I am in Michigan and my Moonbeam coreopsis is about 1/2 inch tall. It is in full sun. The plant dies back to the ground in winter so all above ground stems are sheared in early spring. If you are speaking about one of the larger coreopsis plants, they should be up by now up to zone 4. Look for seedlings as this plant, depending on the variety, self-sows freely. Coreopsis do not like the temperature dips of early spring so they wait until the weather has settled before emerging. If you care to repost with which coreopsis you grow and where you live, we'll try to be more specific.

That can be hard to say without seeing the pest responsible. There are several things that you can do to prevent this from happening, though. Many natural solutions are very effective against a broad range of insects.

This collection of articles will give you ideas on how to make your own pest control, along with many other very useful tips: https://www.gardeningknowhow.com/special/organic

Lance leaf is one of the common leaf shapes and does not refer to a specific plant. It means a narrow leaf that comes to a point like the end of a spear or lance. Some cultivars or hybrids have varying leaf shapes so "lanceleaf" is used as a descriptor to separate it from its cousins. One example is lance leaf coreopsis. Do you have the name of the plant in question?

Where I live, coreopsis dies back to the ground in winter. No pruning is necessary. Maybe I'm not understanding your question. Please do another post and list where you live and what type of coreopsis you have.
https://www.gardeningknowhow.com/ornamental/flowers/coreopsis/coreopsis-overwintering.htm
https://www.gardeningknowhow.com/ornamental/flowers/coreopsis/growing-coreopsis-flowers.htm

This is best done in early spring. They usually reseed each year, so there will be plenty to choose from. If you would like to take a tip cutting, take the tip of the plant before it flowers, and root it in rooting hormone and perlite, or other rooting media.

This article will help you to grow these plants: https://www.gardeningknowhow.com/ornamental/flowers/coreopsis/growing-coreopsis-flowers.htm