I know I shouldn't answer such an unrealistic hypothetical question but here goes...
whatWhat would happen if Jupiter became solid like earthEarth and condensed to the size of Earth?
Well at these densities the transition from gas at the edge to whatever you'd call it at the center of Jupiter is a bit complicated, there aren't distinct gas/liquid and liquid/solid boundaries.
So really Jupiter could not ever be "solid like Earth" and simultaneously "condensed to the size of earth".
...if Jupiter was... the size of earth... would... it's... gravity, orbit, rotation... stay the same, or would it adopt new ones?
Surface gravity is given by $GM/R^2$. With $GM=\text{1.266} \times 10^{17} \text{m}^3/\text{s}^2$ and $R=\text{6.378} \times 10^{6} \text{m}$ it would be 3112 m/s^2 or about 317 times Earth's surface gravity
Its orbit around the Sun would not change much at all; as far as the Sun is concerned it's already essentially a point mass.
Its rotation would certainly speed up! I thought this might come in to relevance some day; Jupiter's angular momentum is about $4.30\times 10^{38}$ kg m$^2$ s$^{-1}$. If we changed the shape of the density profile to match Earth's then we'd use the profile and mass to get a new moment of inertia and use that to get a new rotational period. However, if we kept the same profile shape we can just increase the rotation rate by the inverse square of the change in the radius. It would spin about 120 times faster.
The problem with that is that it would distort into an absurdly oblate spheroid and probably blow up. I say that because if it remained a sphere the centripetal acceleration $R \omega^2$ would be about 2840 m/s$^2$ or nearly equal and opposite to its surface gravity.
So this is an untenable situation and the planet would likely break up.