If the Sun quit producing energy tomorrow, how long would it be before we would notice?

The Sun would start to heat itself by gravitational collapse and by emission of stored energy from the present heat. It would be lit up another 50 million years or so, but in what configuration I do not know. The "equation of state" that governs the distribution of matter and energy within stars would change once the energy source became the gravitational potential.

Is it likely that the reactions in the Sun would go out?

No. If they momentarily faltered they would start right up again.

If our Sun works the same way and is more powerful than many hydrogen bombs, why aren't we burned by radiation?

The energy producing region at the Sun's center is blocked off by many thousands of kilometers of matter. The center of the Sun is not transparent. The energy must percolate upward largely by the process of convection and it tends to lessen from nuclear energies to heat energies. The only form of radiation that emerges from the nuclear reactions at the core are neutrinos, which basically pass right through us.

But if the Sun isn't transparent, why isn't it dark?

Opaque doesn't mean dark; fog isn't necessarily dark, but you can't see through it if it's thick enough. The energy created in the Sun's interior is quickly captured by another piece of the Sun, which then re-emits it in two or three pieces and so forth. Eventually it reaches the photosphere right on the outside of the Sun, where the density of the gasses drops away sharply and the Sun becomes transparent. Half the emitted energy is emitted upward and escapes.

How much energy does the Sun have stored inside of it?

Of course, it depends on the distribution of matter and its temperature at various depths, but some models suggest that the Sun has 30 million years of energy emitted at the current rate banked up inside of it.

I once saw a picture of the Sun covered with dark spots. What causes sunspots?

The Sun has a magnetic field. This magnetic field is trapped inside plasmas, which is what we call hot regions of ionized matter. These magnetic fields are then twisted end over end in the convective regions near the Sun's surface. These twisted regions of high magnetic activity are called sunspots because parts of them are cooler than the surrounding regions (they're still plenty bright, just darker than the rest). The twisted magnetic fields can also break like rubber bands, dumping their energy into the surrounding gas. Such heated gas is called a solar flare.