I've been in the district since 1996 with 1 child all the way through, the other finishing 9th. I'll will try to answer your questions.
IG has been in Mount since I've been here. There's one class for each IG grade level -- dividing it among the home schools would be a disservice to the children. If you haven't read this, it might give you some idea as to how children are selected for IG:
Three Village Central School District
There is an enrichment program, Pi, for those who haven't made the IG cut. Pi is based out of the student's home school. Children are pulled out during the day.
Three Village Central School District
My suggestion for more detailed questions and answers would be to contact one of the moms at 3VSAGE as they will provide you with significant insight:
3VSAGE | Three Village Schools: Advocates for Gifted Education | Wix.com
IG kids have an advantage and disadvantage once they arrive at one of the junior highs: given that some of these students are off-the-charts intelligent leaving them socially awkward, they have a built in support system of IG peers who understand them. They're also moving along in honors or AP classes with some of their Pi friends from their home schools. They're not in a bubble, not isolated, and because of other interests such as music, sports, and the Science Olympiad (Gelinas #1 in NYS this year) many fit in just fine and tend to gravitate to fellow students who like challenges.
With respect to elementary level work -- my youngest is several years removed, so my information is dated. My experience was that the children were given reading materials appropriate to a child's individual level. Writing was also developed at the individual child's pace. After my youngest moved up to junior high, the school brought in lattice math. I do not know whether or not this is still being taught, or if it is a casualty of Common Core transitioning. The elementary schools have science labs, the ones in Setauket were all redone in 2010(?) -- it was fairly recent. I can't honestly answer your K questions as it has been too long. All I can say is there was a lot of hands on work. Overall, I felt the children throughout the elementary did not get as much physical movement type activity as they should have. I was not alone in that opinion.
Many students participate in the music program and quite a number of students go on to NYSMMA. Children start with recorder lessons in 3th grade and move on to other instruments in 4th (if they haven't already begun an instrument at home.) I have been pleased with the music program; both of my children selected brass instruments. There are some wonderful music courses at the HS level; my oldest's former girlfriend was taking composing and some other very interesting advanced level courses in preparation for her studies in music therapy. My oldest was involved in concert band, jazz band, and marching band which in the last couple of years (too late for him) has begun morphing into a competitive, choreographed team. (Trivia: Oscar-nominated composer Marco Beltrami graduated WMHS)
Being that the university is an ever present entity, there are a fair number of families that move to the area for employment and into the district. This results in a good number of bright students from well-educated families. There is a good amount of healthy competition, but during science fair nights one can tell whose parents are SBU research scientists.
One additional positive that the movement of SBU connected residents into the community brings is that many people are from somewhere else -- it's not a vacuum consisting solely of Long Islanders. Our children have made friends with children from all over the US and the world. This results in a friendly environment at the elementary level.
Once hormones kick in heading into JHS, it's a different animal. Social groups and cliques have formed, but there are always students who will take a new student under their wing. A few friends have opted to send their children to St Anthony's, Knox, and Stony Brook School as opposed to attended one of the junior highs and WM. My observation is it is more of either a religious component, sports, a child's behavioral problem, or having the money to send a child to a small, private school.
Last year's budget cuts were painful. Aside from Latin and Mandarin, we also lost German and ASL, which my youngest wanted to take. The theater course at the JHS level were dropped and turned into a club. Students fought to preserve Student Government. I believe a few AP courses were dropped -- ones which historically did not fill a classroom. We did lose some dynamic young teachers, which was a complete shame. This year's budget hasn't raised the fuss last year's did and the troops haven't had to rally to save one program or another.
As for thoughts on neighborhoods, please send me a direct message.