It looks like a weeping cherry. The branches are meant to hang down low. If you cut them back you will ruin the shape of your tree. Google image 'weeping cherry' and see how good a well-grown specimen looks.

To the best of my knowledge, Mimosa Pudica is really just a perennial or at most a small shrub. Since it's not hardy in my zone, that's the only way I have ever seen them. If it's a tree I surely can't think of anything in that zone other than an Albizia with that leaf configuration and with those 'magic' leaves. I don't think I have ever seen any documentation to the ability of the albizia to move its leaves in any other than a light response, but I have noticed when handling them on my own property they do have a slight response to strong touch, but not to the extent of the mimosa pudica nor to the extent they respond to light/dark. They are related, btw.

Dirtaddshp, 5 feet is too close for a mixed planting. It's about right for a solid leyland hedge but a hedge like that either wont be long lived, require regular shearing or all of the above. When trees are planted that close together they will grow narrower but there comes a point where they just get too big and start shading each other out. I would not recommend the arborvite. I know they are very popular in the rest of the country but there are trees that tolerate the heat, drought, and soil better. Another one to avoid is norway spruce or blue spruce. Don't forget to add in some shrubery too.

Do you know if it is a Japonica or a Sasanqua? Did it flower in spring (Jap) or fall (Sas)? Does it have large leaves (Jap) or small leaves (Sas)? If it is a Sasanqua it can take alot of sun. If it is a Japonica, the early morning sun can cause frost damage on the buds, they do best in all day high shade, with adequite moisture. If the plant seems happy in all other regards, I would sprinkle super phosphate into the root zone and work into the soil with a spading fork. If you can't afford to go out and buy some super phosphate try mixing a tablespoon of shampoo (most all soaps are made of phosphates, shampoo is typically the most gentle) into a bucket of water and pour it slowly over the root zone.