if you live in Norwich, then for middle school and high school, your children go to school in Hanover, NH. Hanover and Norwich are in the same school district. those schools have an excellent reputation for sports and academics. not sure how a struggling student would fare, but the school district probably has the most resources of any in the area, as the socio/economic demographics are higher than Hartford, VT or Lebanon, NH.
the web site for the richmond middle school is
Richmond Middle School
i think you will find that most towns have a single high school and middle school, so the choice is not about neighborhoods but about towns and/or school districts. Some towns have no high school. Lyme, NH sends their kids to Hanover High. Hartland, VT may send some but not all there.
Size of Hanover High School is about 750 to 800 students. not sure where that fits size-wise.
BTW, White river junction is actually part of the town of Hartford, VT
good luck.