Here are my thoughts. All the places you listed are safe and you will find "down to earth" people in every community (though I wealthier communities MAY have less down-to-earth people).
1) Scarsdale - great town, excellent schools but there may be a "keeping up with the joneses" attitdue. The area by the Edgewood school district is the more "affordable" area of Scarsdale, nice community, friendly people. The "keeping up" attitude may not be present in that community but will be when your kids go past elementary school and are mixed in with kids who come from wealthier areas of Scarsdale.
2) Bronxville - I love Bronville. It has a beautiful town center and excellent schools but the homes are very expensive and the taxes are very high (taxes in Westchester are high in general but Bronville taxes seem to be on the highest end)
3) Chappaqua - haven't looked at homes in Chappaqua for awhile but imagine they are also very expensive.
4) Rye - I think Rye "proper" ranks right up there with Chappaqua. Million dollar homes but also teeny tiny overpriced homes near Playland Park. For $820K, you may still be looking at a fixer upper.
5)Larchmont - I love the village of Larchmont. Great community and there is a beautiful park there (Manor Park) that I visit often. Just note that Larchmont shares a highschool with Mamaroneck. I don't know too much about Mamaroneck schools (I do know they are not bad per se but I don't know if they are considered good either) but I would look into that before purchasing in Larchmont.
6) Pelham - nice town, diverse, close to a train line, great schools. Surrounding areas of Mt. Vernon and New Rochelle are not so great in my opinion.
7) Tarrytown - diverse area, great town center but schools are not as good as all the ones mentioned above.
8) Pleasantville - very nice town, good schools - don't know as much about Pleasantville as other communities so can't comment too much here
9) Hastings - I personally love Hastings. We almost lived there. In fact we looked in Larchmont, Hastings and Ardsley. It has a decent town (not as nice as Larchmont and Bronxville) but not terrible either. Excellent schools and a wonderful community. I would recommend Hastings in a heartbeat.
10) Harrison - Great community, taxes on the lower end, nice town. Can't go wrong with harrison, This month's Westchester Family had a write up about Living in Harrison.
11) New Rochelle - schools not as good as the ones listed above, diverse, not a good town center. There are areas of NR that are better than others and there are areas that are downright scary. Another thing to consider about New Rochelle is the re-sale value of the homes. Homes don't appreciate as much as other they do in other areas of Westchester.
12) Ardsley - great schools, diverse, but the town center not nice. I live in Ardsley and I love it. Its a wonderful place to raise a family, the people are friendly, lots of recreational activities. $820 can get you a decent house.
13) White Plains - more city-like than the other areas, schools are decent. I wouldn't recommend going from NYC to White Plains if you're looking for a suburban feel.
You didn't mention if you need to be near a Metro North Station. If so, I would look at towns that have easy access to a Metro North station.
Of the ones towns you listed I would look into Scarsdale (Edgewood Section), Hastings on Hudson, Ardsley and Larchmont.
Here are some sites that might help
Houlihan Lawrence
Houlihan Lawrence Real Estate
Westchester Board of Realtors (don't have to be a realtor to browse homes)
Welcome to WCBR.net!!!
Journal News - links if you want to look up school district stats (under education)
lohud.com | Westchester, Rockland and Putnam Counties Data Central | The Journal News
Good Luck. Hope this was helpful.