What's the size needed for cracking large-primes based encryption? If not at 100 qubits, what are some useful calculations that we can do at that size?
That's still many times faster than you'd get conventionally, so it's a reasonable trade off.
But it doesn't address the possibility that you can't entangle enough qubits to begin with, or keep them stable long enough to actually perform the calculations, which is where the error correction stuff comes in. I don't know enough about the challenges there to comment on how many you'd need (though 1000:1 seems high to my naive intuition, i'd guess maybe 10-100 for magnitude myself).
newbie question : are they naturally instable (so we won't fix that) or is it just because we don't master them well enough right now (and we'll fix them "soon") ?
However, with the right kind of quantum computer or quantum simulator, you could construct a system of qubits that is described by the exact same Hamiltonian. That way, the quantum state of your qubits and your original system would behave in exactly the same way. Then, you just let the qubits evolve in time and read out the system's state at the end. Do that a bunch of times and you'll see an average picture of what the original system you're modelling (protein or something) would do.
So to recap - we can get around the difficult compute bottleneck caused by the desire to perform high-fidelity physics simulations by creating a system that follows the same rules and which we can probe much more easily.
Your question doesn't make sense. Many (most) things in nature are incredibly hard to model correctly. Just because they happen doesn't make it easy to quantify usefully.
For other calculations you might need error correction, but this leads to nowhere with current technology. What the Chinese did was groundbreaking.