Theoretically, if an asteroid was dense enough, could it assume hydrostatic equilibrium

I mean if some space rock was only like a mile wide, but it was super dense, could it be rounded under its own gravity?

Not unless the asteroid was made of same magical material. If it was made of solid osmium (the densest ordinary material known), the asteroid would have to be 253 km wide (157 miles).

Basically, you need a mass of 240 x 10^18 kg, of anything that can crush under it's own gravity.

Mimas is mostly ice, which crushes rather easily compared with rock or iron. For a ice asteroid, you still need a diameter of around 370 km to make it round. Mimas is 396 km, so it just barely makes the limit.

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In addition, Mimas gets tidal flexing from Saturn, and the Hershel impact would have mostly melted it. Liquid water forms a sphere very quickly, at just about any size!

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That's what happened with Ceres. The collapse due to a hydrostatic equilibrium releases a lot of heat allowing for the material to differentiate and form a core which starts off molten but cools quickly because of the size of the planetoid. I don't think a mile would do it even if it was very dense. There are irregular shaped asteroids larger than a mile across.

Were the one mile wide asteroid made completely of iridium it would have a surface gravity of 9.7 mm/s². This is about the surface gravity of the considerably larger asteroid Ida which is nowhere near spherical. Instead of density, concentrate on making it out of a low viscosity, low elasticity material. A custard cream asteroid would eventually pull itself into a sphere. Use a heavy cream.

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Yes, it could, but it wouldn't have an atmosphere so it would still be completely covered in craters. It would probably be colliding with many rocks because it would have quite the gravity. After a while, I think it could look somewhat like the moon.

With density usually comes mechanical strength.
At a 1 Mile scale most solids could resist deformation enough to prevent it.
Now it your asteroid was made of mercury????

yes. We see this happening with drops of water on the space shuttle and the ISS, just a few inches across.

yes