Homeschooling is no longer illegal in any state; however, different states have various laws regulating how homeschooling is carried out.
As it happens, there is a non-profit organization that protects the rights of homeschoolers. It is the Homeschool Legal Defense Association, with a website that even color codes the different states. Here is the link, if you want to find your own state:
No ... in fact, homeschooling is protected by federal regulations ... which state, among other things, that school districts have to allow homeschoolers the same opportunity to participate in extra-curricular activities as "regular" district students.