Hi, 

Those are just root suckers that are coming up at the base. It is very common with Crapemyrtle trees. You should prune those off before they get too big. They will probably re-sprout so you'll just need to keep doing it. 

The white covering looks like powdery mildew. If you don't see anything up on the main branches, then by cutting off the shoots that should be sufficient at controlling it. 

Sometimes trees will send up a lot of root suckers when they are being stressed for another reason or there was root damage, so you may want to try to examine the upper branches and leaves carefully. We are getting more instances of Crapemyrtle aphid and Crapemyrtle Bark Scale and have seen a lot of presence this spring. 

You can also try to send us in photos of the crown and canopy of the tree to see  if we can notice anything. Keep the tree watered as we have had drought like conditions this spring and even with recent rainy weather we are below average for rainfall still so will see more stress in our landscape plants, even mature trees. 

Emily