Strictly speaking, the First Amendment says that CONGRESS shall make no law. So, the states (per the 10th) could establish some restrictions. But, by convention, it was decided long ago, that the First Amendment covered all government: Federal, state, local.
Oddly enough, the Second Amendment does not mention Congress. The Second is a sweeping statement that covers everything imaginable and says that the right to keep and bear arms shall not be infringed. By anyone. And, somehow, it was decided a long time ago that the Feds can infringe, the states can infringe, and the local governments can infringe.
Good point regarding the 1A. Agree. Scotus unconstitutionally incorporated the Bill of Rights.
I imagine that if someone is using or trying to use their "right to keep and bear arms" to infringe on my right to life, I, as "anyone", will infringe on their right to keep and bear arms.
So many stress "shall not be infringed" without adequate attention to "what" shall not be infringed, said what being a right. So start by defining the noun "right".