Is this algorithm for simulating a quantum computer accurate?
Is this algorithm for simulating a quantum computer accurate?

Yes, this will essentially work, although there are some pitfalls in the way you've phrased things. For one, you cannot store each qubit as a single bit, but you can work things out so that any computation you do starts from a state which can be represented by $N$ classical bits, i.e. the zero state.

Following this your second and third bullet points are correct.

As for the fourth, element-wise squaring implies that your measurement is in the computational basis. If you need to measure in, say, the Fourier basis, you will need to convert to a computational basis measurement by including a Fourier transformation in step 3.

Bullet 5, like bullet 1, conflates the classical information encoded onto/extracted from qubits with the qubits themselves. Everything here is fine assuming you mean for step 5 to be the absolute end of the computation, but you cannot, in general, loop from step 5 back to step 2.

Ultimately the reason your simulation works is that any computation on $N$ qubits can be represented as a single ($2^N\times 2^N$) unitary acting on the zero state followed by measurement in the computational basis. You cannot, however, use a simulation such as yours to do gate-by-gate simulation of a quantum algorithm.
    