I would not advise using round-up on the sprouts. If they are sprouts that came up by seed away from the tree, it wouldn t hurt. Most of the sprouts are root sprouts and are attached to the roots of your main plant. Glyphosate products such as Round-up, work by entering the foliage and then move down to the root system. While the spray wouldn t kill a large plant, you could have some damage.
Sprouts are a fact of life with crape myrtles.
Heavy mulching can reduce it, but just keeping them cut or mowed should suffice. It doesn t have powdery mildew for once, but the trunk is shedding. There are huge sheaths of thin bark which come off in layers of scales. Your tree is perfectly normal, it has simply come of age. Crape myrtle trees shed old bark as they age, leaving behind wonderful shades of tans, grays and browns.