Not at all! I've actually been surprised at how far/long i can run with music and the app on without it draining my battery. i have an iphone, and i turn my screen off when I'm not actively using it during the run.
I do. The phone can only perform as well as your cell signal will allow. If it has to work harder, you will drain your battery faster. Also, make sure all other apps are closed... even use a different device for music.