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Well it's not wet, but if you stopped the reactions, the sheer force of gravity of the upper layers on the inner layers would start them again. Not that that has any effect on its mass. (I mean it does, because the nuclear reactions convert mass to energy, but that's a very long process.)
Asking weight doesn't really make sense, because weight is a product of gravity. The sun has a mass of 1.9891x10^30 kg, and at 1g that's 4.384x1030 lb, at least according to the Google result summary I copy-pasted from.
How much would the earth weigh if it was sitting on the frozen surface of the switched off Sun?
The Earth's weight is 1.317×10^25 lbs. Gravity on the "surface" of the sun is about 28 times Earth's, so if the Earth was a point mass (it isn't) on the surface of the sun (there isn't one, it's a soup of plasma 250 mi deep) the Earth would weigh 3.687×10^26 lbs.
Of course that couldn't happen anyway because part of the reason the sun is the volume it is, is because of the nuclear reactions blowing it apart. If you stopped the sun, it would collapse (and that crushing energy would restart the reactions).
Also the earth isn't a point mass, so the point where the Earth and Sun are touching would experience more gravity than the point on the opposite side of the earth.
I'm sure it would be the same as it currently does.
Weight depends on location. For example, you would weigh less on the moon.