I can't tell you anything about schools, but I lived in the borough of Croydon for five years. I loved it.
Thornton Heath is grim, but the hospital is there.
Purley, Kenley, Coulsdon are all nice areas. Even South Croydon isn't too bad. Every single London borough has crime problems. Croydon has a knife crime problem, but then so does Lambeth and Southwark. Violence and sexual offences is highest in the City of Westminster. Robbery in Haringey in 4 per 1000 people, compared to just over 2 per 1000 in Croydon. It's a bit swings and roundabouts. As a woman walking around on my own day or night, I never felt unsafe in Croydon whether it was Croydon town centre, or some of the dark roads near my home.
Croydon is what you make it. There is so much to do there - you're never bored. There are great places to eat, some great places for entertainment, always something to do at the weekend. You never have to travel far to find something. I honestly don't think I ever left the borough in order to buy something even if it was a big purchase. The Purley Way is immense.
It being South London, the transport links into Central London are very good, depending on where you end up (where I was the trains were every 15-30 minutes) you could have just mainline railway or you could have the Overground service too, and there are tons of buses.
If I could move back to Croydon tomorrow I seriously would.
Hope you find somewhere to suit you! 