How dependent is Earth is with the other Planets' gravity?
The orbital speed of the Moon around the Earth is 1022 m/s. The Earth is 81 times more massive, than the Moon. Thus, a sudden disappearance of the Moon would cause an $\frac{1022}{81}=12 \frac{m}{s}$ change of the Earth's orbit, which would be negligible.

The disappearance of the Mars or Jupiter would have yet lesser effect.