Chiming in from Texas. We're a "just do it" state. Nothing to file, no requirements, and nobody to inform.
Regarding participating in public school activities, it is up to each individual principal, but I've never heard of anyone turned down. The thing is, we have so many options here for extra-curriculars for home schoolers, I'm not sure you'd really end up being interested in the public school activities.
Texas has so many homeschoolers (the most in the nation), that you'll find daytime homeschool Co-ops, athletic leagues, service organizations and private classes like dance, cheer, martial arts, art, etc. in the major cities.