Probably but it would take a while. Any relative angular momentum between the Sun and the black hole would result in the hole orbiting the Sun since an earth mass is tiny compared to the Sun. It might well orbit beneath the surface of the Sun, sucking up plasma as it orbits. How long would it take to "eat" the Sun? The calculation could be done but it's tricky; the hole grows as the Sun shrinks, and that causes the mass transfer to accelerate. And the Sun is denser in the middle and the hole will spiral inward as it grows to keep the center of mass of the system in the same place, etc etc.

It's possible there are one or more black holes orbiting inside the Sun right now. They might be the reason sunspots exist.

Depends. Is the velocity of the black hole relative to the sun higher or lower than the escape velocity of the sun?

If higher than it's going to keep going, if lower, then yes eventually it's going to consume the sun.

The sun emits earth sized mass all the time, I imagine the blackhole could get sway to different directions.