What would happen if 1 cm^3 of matter from the center of the sun instantly appeared at the surface of the earth?

Not much.

No fusion or anything (once isolated, at Earth pressure, it would cool down).

So the main effect would be the transfer of heat energy.

That 150 g of hydrogen at 15 millions degrees. So 0.15 × 15×10⁶ × 14×10³ = 31.5 GJ

That is 7 tons of TNT. It is a big kaboom (the infamous car bomb of Beyrouth was only 1 ton of TNT).

So, I hope it doesn't happen in my backyard.

But further than that, no real consequences I would say.

There is also the sudden expansion. That 1cm³ will want to occupy some 1.7 m³ once on Earth. (Far more in reality, because it is very hot. But if you count that it is more, then you must also count that expansion would cool it down. We don't want to count the same energy twice). But that energy is negligible (some kWh), compared to the first one.

I sure use some very naive computation (adiabatic decompression of perfect gas, simple calorific transfer, etc.).

With those huge pressures and temperature, none of those rules are true.

Nevertheless, it give an order of magnitude. At least an approximation of it (I mean, maybe it is 100 times more, maybe it is 100 times less. But in both case, it would be a noticeable kaboom, but it wouldn't destroy Earth, not even the city in which it appears)

The Earth would implode ....

It would bore a hole, pulled by gravity toward Earth's core, but would cool before reaching the core and become part of the Earth. 

It would probably be relatively benign. The fusion reaction would cease instantly since it would instantly expand beyond the density required to sustain fusion.

Remember Hiroshima?