This is almost certainly Earth Star Voyager. It has the crew with children:
Command cadet Jonathan Hays, twenty-one, and his best friend Jessie 'Beanie' Bienstock, a 14-year-old computer specialist, are among the young military cadets chosen to serve as the crew of Earth*Star Voyager, the planet's newest interstellar vessel.
The captain is blown out of an airlock and a witnessing child accepts the responsibility for taking over:
While checking the repairs on an airlock, Captain Forbes finds himself trapped inside with the system set to open the airlock to space. Moments later, coming to check on the Captain, Jonathan finds him before the airlock opens. Before being blown into space, the Captain tells Jonathan to keep going and to 'complete the mission'.
And they pick up two guys from space stations although the one from an abandoned one turns out to be very helpful. It's the second guy that's a pain.
Along the way to Demeter, they pick up the long lost astronaut Capt. Jacob 'Jake' Brown, while stopping at an abandoned station to make some repairs. Brown initially paralyzes Huxley with a weapon but it is revealed that he did so to prevent Huxley from walking straight into an "anti-matter zone."
The next discovery, explaining how Vance and Willie got to the World's Fair Station in the first place was finding the wreckage of the Vanguard Explorer ... [and] ... an unwelcome guest known as a 'Shell'—a cyborg. Brown is able to shock the Shell into unconsciousness, and Doctor Arthur insists they take him to the ship to help and examine him...
... After making some small adjustments to him to ease his discomfort, Doctor Arthur and he form a brief emotional connection before his programming returns him to his drone-state. Later he regains his full movement and smashes his way through the ship, damaging Capt. Brown's jury-rigged railgun weapon and one of Priscilla's processors. The Shell warns them that they must stop "Assembly" then deactivates himself rather than let his programming make him kill Dr. Arthur.
You can generally find copies of it on Youtube. It was a miniseries never released on DVD although there was at least one VHS copy.