Corporations are indeed creatures of government. Are they required to conduct business? No, but they are awfully convenient, and allow for a lot of rotten things to occur.
Private property? If it requires government's nod, it isn't private. And, of course, we see this all the time, when government decides that it wants someone's property, and takes it. It might be your land, your automobile, or something else. It is all up for grabs.
Not sure why you're name-dropping Rothbard, but it doesn't help you here. No business is going to exist without participating in the existing society. Trade and commerce require cooperation. The alternative is what brings a society closer to subsistence farming (and it tends to leave people starving).
You can't escape your problems by pretending to define them away. No property regime has ever been secured without the use of government.
We've had a chance to see a number of government experiments run their course (some continue to), ranging from absolute monarchies, to Marxist-Leninists, to Maoists, to various kinds of republics. All of them have ended up going in similar directions, over time, consolidating power, though at different speeds.
You can tell yourself whatever Rothbardian fables you like, but the only real business you're building without any participation in a modern, "coercive" society is a subsistence farm.