How soon can we start planting annuals in zone 4b
You will need to wait until the danger of frost has passed.
As a zone 4b garden myself, I know that this is a date that changes and you need to monitor the weather, sometimes even into mid May we can see a frost.
May 15 is the safe date. If you are potting up annuals you can often do so earlier but may need to cover or move indoors if our weather takes a dip.
Only directions show it plopped into a container still rolled up. I want to unroll. One side has white mat, other has green shreds.
Plant the seeds up. You may have to cover it with soil.
Here's an article about how to make your own:
https://www.naturesseed.com/blog/how-to-make-plantable-seed-paper/
To plant
You have many choices! These articles will offer choices in, both, annual and perennial in case you wanted to expand your potential choice range:
https://www.gardeningknowhow.com/garden-how-to/gardening-by-zone/zone-6/plants-for-zone-6.htm
https://www.gardeningknowhow.com/ornamental/flowers/fgen/annual-plants-for-cold-climates.htm
https://www.gardeningknowhow.com/garden-how-to/gardening-by-zone/zone-6/zone-6-flowers.htm
Sunbathers, etc killed by hard freeze. Do I remove or mulch?
It's generally better to pull out dead annual plants rather than tilling them in. Pulling them out helps remove the plant material from the soil surface, preventing potential diseases and pests associated with decaying plant matter. After removing the plants, you can add a layer of mulch to help suppress weeds, retain moisture, and improve soil health. Tilling can disrupt the soil structure and may bring weed seeds to the surface, leading to more weed growth. Mulching over pulled plants provides a cleaner and more effective way to nurture your garden soil.
can annuals be stored over winter and or harvest seeds and how?
You will have to save the seeds. This article does a really good job of explaining more:
https://www.gardeningknowhow.com/ornamental/flowers/fgen/harvesting-garden-seeds.htm