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You do not need to dig it up. It will die back and, in most cases, pop back up next season - though it may be elsewhere through reseeding. You can always collect seeds for replanting in spring as well. Here is more info on this plant: https://www.gardeningknowhow.com/ornamental/flowers/forget-me-nots/growing-forget-me-nots.htm

Yes, you can plant Forget-Me-Not's among your Hosta, but I will caution you that these little delicate flowers can become invasive.

Here is an article with more information.

https://www.gardeningknowhow.com/ornamental/flowers/forget-me-nots/controlling-forget-me-nots.htm

Yes, forget me nots can be planted under pine trees. For more information on forget me nots, please visit the following link:
https://www.gardeningknowhow.com/ornamental/flowers/forget-me-nots/growing-forget-me-nots.htm

You can trim off the spent blossoms with a scissors if you would like to clean them up.

Here is a link about the care requirements.
https://www.gardeningknowhow.com/ornamental/flowers/forget-me-nots/growing-forget-me-nots.htm

Forget-me-nots are winter hardy down to zones 3 or 4, depending on the variety, so they can survive outdoors in the ground in Minnesota, although in some regions they may need to be protected by placing mulch over the soil. They can be transplanted either in fall or spring. Choose a cool, preferably overcast day to transplant, and follow this advice for preventing transplanting problems:

https://www.gardeningknowhow.com/plant-problems/environmental/learn-how-to-avoid-and-repair-transplant-shock-in-plants.htm

Be sure the soil is constantly kept moist, but not too wet. Seeds will germinate best if they are kept constantly moist and never allowed to dry out. Forget-me-not seeds take 8-14 days to germinate, so you may still see them in the next few days. If not, you could try soaking the next batch of seeds overnight before you plant them. And if they still don't germinate, the seeds might be old or poor quality, and it would be best to get seeds from a different source.