Factual error: When the Jupiter 2 blasts off, it does so in the most fuel inefficient way possible - flat surface first. The air resistance would be fifty times less if they'd angled it side on. Plus, when the aerodynamic cover blasts away, the Jupiter 2 itself is revealed - and it is more aerodynamic than the cover. Rocket science must have taken a few steps backward lately... .
Plot hole: Considering that this family has been training for the mission for years and certainly have received some instruction in astrobiology, they are awfully carefree around the cute little alien. Any real space farer would be aware of the enormous potential problems with first contact, like incompatible biochemistry (aliens might exude deadly toxins from their skin, be eaten alive by human-exhaled oxygen, etc). They even feed it human food without analyzing the alien; again, this could kill it, make it chemically dependent, cause it to go into a mating cycle, etc. But it's cute alien, so naturally they don't worry about any of this.
Plot hole: At the end there were so many. Not having enough time to get through the center and out the other side, the fact that gravity would pull them back toward the center. Also, a planet cannot become a black hole despite the added energy unless it was an implosion to the center. They also needed the core material for the hyperdrive as well.