If by 30-45 minutes you mean door to door, that's not going to happen. ~1 hour is more realistic and could be substantially more. Even Pelham, which borders the Bronx, is a 30 minute train ride. Add in travel to time to the station, a couple minutes of buffer time so you don't miss the train and then however long it takes to get to your offices you might be able to make that happen in just around 45 and that's if you work right above or next to Grand Central.
For schools, peruse
this list and sort by SAT score. That will give you a pretty good idea of the pecking order.
The overwhelming majority of people in the places you'll be looking will be white, Judeo-Christian (some non-believers), heterosexual married couples. How you are received will depend on what sort of minority you are. For example, if you are a gay American Jewish couple, I think you'd have a much better experience than an observant immigrant Muslim family.
You are looking in the right places, but you aren't going to fulfill all of those things at once. I think Pelham probably comes the closest (lacking in the diversity department), since you are OK with apartments means you can live near the main drag and walk to the station. The rivertowns are very close behind. Riverdale also comes into play, it is better in the diversity department, but the schools are probably worse. Plus NYC tax.