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I am not sure...what zone do you live in? For more info on growing the plant, this article will help: https://www.gardeningknowhow.com/ornamental/flowers/lungwort/growing-lungwort-flower.htm

No, wait until winter to cut back the woody stems for spring regrowth.

https://www.gardeningknowhow.com/edible/herbs/wormwood/wormwood-plant-growing-sweet-annie.htm

You can prune back dead or old foliage at any time, and you can remove any that is spreading outside the area you want it in at any time. It's also good idea to deadhead (remove the spent flowers) after blooming.

It is definitely fungal. You can likely correct this by mixing 1/4 peroxide to 3/4 water next time you water it. This will only need to be done once, or any time that it is needed in the future.

Just make sure that the top 2 or 3 inches are completely dry between waterings. This is likely how the fungal infection came about, by being too wet for too long.

As far as the weeds, the largest is a dandelion, the other two have many lookalikes, and will be harder to distinguish at this age.

These plants don't thrive in full sun, as you're suspecting. Here's a helpful article:
https://www.gardeningknowhow.com/ornamental/flowers/lungwort/growing-lungwort-flower.htm

If this is stinky because it has gone rancid... I wouldn't recommend feeding it to your plants. It could cause them an infection.

If it is fresh, and you find the the smell, particularly, unpleasant and you know that it is still good, I'd say it wouldn't harm the plant.

The worst that could happen is that you could find some growing on the soil surface.