We can try to partition the machine so your process can use some qubits and mine uses others. I suppose a classical computer would be running the OS, at least at first.
There will always need to be a classical computer running the quantum computer.
Not sure about the need, but it sure is convenient. Quantum computers are not always better for all kinds of problems and being able to route different jobs to different parts of the system should be an advantage. All this looks a lot like a digitally controlled analog computer or something that can program FPGAs on-the-fly.
One big limitation of quantum computers is that they can't erase information without losing their quantumness. So simple things like if-then statements become impossible (you'd have to execute both branches every time).