I'm on board with your first two proposals, but there is no way you can require private businesses (even if they are open to the public) to allow people to bring guns into those establishments. Aren't you a big free market/libertarian guy? If I recall correctly (and my apologies if I am getting you mixed up with another poster), you think privately owned businesses should be able to deny services to customers based on whatever (i.e., Christian baker should be able to refuse to make wedding cake for gay wedding based on religion). If I am recalling this correctly, how do you reconcile those positions?
I am a Libertarian but that does not mean I would be against requiring businesses to follow certain regulations in order to do business. It would seem to me the power to enforce one would be similar to the power to enforce the other.
In theory, you could argue that all people and businesses should be allowed to do whatever they want so long as they do not harm others, but we are miles away from that.