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It is likely lacking in phosphorus, which it needs to bloom. Give it some bone meal to add some phosphorus to its soil.

They are seed pods. You can just remove them from the plant. They will only sap energy from the plant.

They will not harm the plant. It is actually a good sign that the soil there is good. This article will give you some suggestions as to how to get rid of them, but if you don't mind them growing there, they will not hurt anything.
https://www.gardeningknowhow.com/lawn-care/lgen/eliminate-mushrooms-in-your-lawn.htm

In your area, I would probably wait until fall unless you can ensure that it receives plenty of water while it re-establishes its roots, which will be important during very warm weather in summer. When you move it, make sure to get as much of the root system as possible and surrounding soil. It will also help to prune it back some so the plant can put its energy into the roots rather than foliage. This article should hlep:https://www.gardeningknowhow.com/plant-problems/environmental/learn-how-to-avoid-and-repair-transplant-shock-in-plants.htm

Yes, it is likely suffering from shock but should bounce back once it acclimates. And yes, the seed pods are in the spent blooms. However, the plant can also be easily propagated through cuttings.

It's likely too much nitrogen fertilizer, which results in more foliage growth and little to no flowering. This can be offset by adding more phosphorus to the soil, either with a high phosphorus fertilizer or bone meal.