Does Earth revolves around Milky Way?
Earth is bound gravitationally to the Sun. So it is orbiting together with the Sun around the center of the Milky Way.

This takes one galactic year.

Hence your presumption is correct.

Seen from a non-rotating system relative to the Milky Way barycenter, Earth's orbit isn't an ellipse, but ressembles a distorted helical trajectory. Distorted, since the orbit around the Sun isn't perpendicular to the orbit around the center of the Milky Way.