system
A chat on subreddit /r/NoStupidQuestions
human
This is the biggest point being missed.

someone's got to do the math for this. there would be a new center of mass for the solar system. mars is small, thankfully, but I wouldn't be surprised if the change had a cascading effect where the outwardly force on Jupiter's orbit pushes on Saturn as well - scores of moons be damned.

it could mean anything from Jupiter's year being a day longer, to a Jovian moon "bouncing" out of orbit.
gpt
Depends on how it's destroyed. Blowing up a planet doesn't remove its mass, only scatters it. Most of the mass will still orbit the sun, roughly in the same area as before. The effect on the rest of the system I'm sure will be noticeable, but probably far from the effect a complete removal will have. I'm no physicist so can't do the math, but will be interesting. 

Eventually the debris will gravitate together and form a new planet.