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The best information I can find in researching your problem with this invasive plant is to use a selective herbicide and treat and retreat.
Many people recommend pulling up as much as you can be hand, then treat the areas with herbicide.
Some advice I found said to use a thatcher on the lawn, rake up and dispose of all the material removed by thatching and then treat with herbicide.
I would suggest talking with your County Extension Office. They may have recommendations on a specific herbicide that works in your region.

https://www.gardeningknowhow.com/extension-search

There are multiple tree species called mimosa; I will assume you are referring to the beautiful Persian silk tree, Albizia julibrissin.
There is no easy answer, especially without photos of tree and site and knowing more about the growing conditions, soil and water management. How long you have noticed this condition, and are there any other signs of disorder such as foliage discoloration thinning and branch die-back?

There is a serious soil borne fungal disease possible, "mimosa wilt". But it would cause wilting and dieback as well as the flowering disorder.

Water management is a key, if one side of the tree gets more water than the other the side with water deficit will suffer.
Drip irrigation systems often distribute water unevenly in the root zone.
Silk trees should be deep watered infrequently and allowed to dry out in between. They require well drained soil.
Root damage, root rot, restrictions to root growth are other guesses. Nutrient deficiencies or excesses could be a reason, if you don't fertilize adequately or if you overfertilize.

https://www.gardeningknowhow.com/ornamental/trees/mimosa/growing-silk-trees.htm

There are plant growth regulator products that professionals use to cause premature flower drop. They are usually used for nuisance fruit. Florel, Oive Stop are two popular products. I don't know if they will work for Mimosa - Silk tree. I frankly have never had a request or known anyone who wanted to do this.
They flowers are messy I know, but I would consider it a blessing and not a burden, they are so beautiful.

Yes this tree will thrive wonderfully, and can even be considered invasive in your area. Here is a link to a compilation of articles that will tell you everything you need to know: https://www.gardeningknowhow.com/ornamental/trees/mimosa/

A photo would help, for more specific recommendations. But the general answer is that yes you can reduce the length of those longer lower branches to bring them into shape with the rest of the crown. Cut back to a lateral branch, don't leave a stub. And don't take more than necessary, 10% of the length would be okay, or 20% maximum at one time.

Absolutely not! This will result in the quick death of the tree. These trees are highly invasive here in Tennessee, but this is not the appropriate time to transplant. This is especially true with the VERY HOT weather we have been experiencing lately.

I would recommend waiting until next spring to do so. This will ensure the best chance of survival after transplant.

This article will give you more information on their care: https://www.gardeningknowhow.com/ornamental/trees/mimosa/growing-silk-trees.htm