Gamma Quadrant? So why hasn't the Dominion taken this place over?
"To the inhabitants of Gamma Germanicus 7, it is the year 2008. More importantly for the students of Hogwarts, it was graduation day. " This sentence jumps from present to past tense. Either is okay, but pick one.
"Will, I'm sensing a strong emotional imprint in that area," Counselor Deanna Troi-Riker said. "I would describe it as nervousness, tinged with anticipation."
She's picking up the feelings of a small group of students out of an area populated with all sorts of people?
"(and a full Betazoid, unlike Deanna) nodded. "I feel it too, sir. The feeling is coming from a castle in the center of the area where the technology was detected. Since I am a full Betazoid"
You don't need to tell us twice that she's a full Betazoid (it sets off the Sue-dar), just work it into the conversation. Maybe she could say, "And I'm picking up anticipation from that big building near the village," after Troi talks of her own findings.
Would Harry really want to get married before he'd found a job and his own place or even completed his auror training? You should address the "I know I'm young, but" factor.
Some of your OC's appear to have high Mary Sue potential, particularly Kandagawa.
On a purely technical note, you don't need to underline the name of a ship. Just italics is fine and proper.
Hysterical premise.