Grab a teaspoon of neutron star core. Teleport it into Earth's atmosphere. What would happen?

Your teaspoon would probably melt and the star would spill everywhere. Get the paper towels.

You're right on most accounts. Imagine Mt. Everest compressed into a teaspoon and then suddenly released to re-expand all at once. That's a fair approximation of what would happen. It is going to expand as fast as the atmosphere will allow it, meaning it will superheat the atmosphere. It will likely result in all free water on the planet evaporating. The side of the Earth where it happens would likely be entirely molten. Given it will expand outward more or less equally, it would also likely eject some portion of the atmosphere into space. That also means there is likely a giant crater, but the molten surface will smooth that over in time - just like how when the Moon was formed from another large object smashing into young Earth we don't have a moon sized crater on the surface. None of this even considers the damage the global shockwave from the sudden expansion would cause. Mostly because everything will likely be gone already.

Why? Because the only thing making a neutron star a neutron star is its gravity. Take a portion of it away from that star, it will explosively expand. Remember, this is so heavy that protons and electrons have been ripped away from their atoms and smashed together to form neutrons.

I'm not a physicist but I'll give it a try...

Assuming you transported it into Earth's atmosphere at a height of 20 kilometers, it would weigh 10 x 10^18 kilograms and accelerate towards the Earth at 9.8 m/s/s (the force of gravity on Earth) which would strike the Earth with 9.8 x 10^18 Newtons of force or about 1 trillion megatons. For reference, the largest thermo-nuclear man-made explosion was the Tsar-Bomba which was a 50-58 megaton explosion. It would result in a crater that's about 120 kilometers in diameter. For reference, the crater that killed the dinosaurs is 150 kilometers in diameter.

The higher up in the atmosphere you dropped it, the more devastating it would be. I don't think you can really consider it to be "atmosphere" much after about 80 kilometers. At 80 km the impact crater would be around 180 kilometers in diameter.

Basically, It would be extremely devastating to life on Earth and probably cause a mass extinction event similar to the dinosaurs.

I dunno if all my math was correct there, but it was fun to try to figure it out.

it will not be an explosion per se but more of an intense neutron radiation https://en.wikipedia.org/wiki/Neutron_radiation

It is a health hazard because when it hits atoms it could potentially make it radio active. https://en.wikipedia.org/wiki/Neutron_radiation