These large caterpillars may be left alone if they aren't threatening a valuable plant. If they are eating too many of the leaves, then kill them if you want to.
Yes, you can do that with a specific insecticide like piretroids or with some product with a similar active ingredient, in the plaguicides' store you can find a few products
The catalpa sphinx worm should be erradicated from your trees. The catalpa worm feeds on the leaves sometimes completely stripping the tree of its foliage. It prefers the Catalpa tree but will infest other species of shade and oramental trees.
