Most of the time, the 'weight' flows back and forth between a and b according to certain equations over time. When you measure the system- that is, when the bit interacts with the outside world, hopefully your measuring apparatus- you see a 1 or a 0, with probabilities |a^2| and |b^2| respectively.

So what you can do is get a whole bunch of these quantum bits- qubits together, and set things up so that the time-evolution of their quantum state is correlated and probabilistically moves towards something you're interested in. Say you can set things up so the bit array- which, at first, will give you a mere perfectly random bit string on measurement- becomes more and more likely to give you, say, a prime factor, or the answer to some other question.

So yes, the quantum phenomenon is that the bits of the computer are quantum objects as opposed to classical.

Unless you're talking about the government putting its own interests above the people's, which makes sense.

Humans are omnivorous; we're presumably adapted for some amount of animal protein. Even if meat-eating goes out of style, once there's no moral hazard involved we'll gravitate back to it.