South Merton is Thameslink - I've had problems with it in the past (the 06:58 being cancelled regularly) but it's useful when the tube is down. It's also the slow train into the city - it takes ages. Morden tube is fine, and you always get a seat which is the advantage of being at the end of the line, though coming home from the city it's packed & you'll probably have to stand to Clapham South/Balham. I've never felt unsafe at Morden but I'm usually there at peak commuter time!
As regards schools, I'm more familiar with girls' faith schools - Ursuline Wimbledon and Holy Cross in New Malden which both give excellent pastoral care, I've found. Ricards Lodge (girls) and Rutlish (boys) - my friends who have children there are happy with them. I'm afraid I don't know anything about the Harris Academies which are the mixed sex schools, other than the heads respond very quickly to complaints about behaviour! All the primary schools are well rated by Ofsted, again I know parents with children at Wimbledon Chase and Poplar who like them - primaries change vibe with the head I've found though.
Epsom is a slightly larger shopping centre, Kingston is BIG, but Wimbledon is good enough although the Debenhams has just closed.
Plenty places to visit - we always like a walk on the commons (Wimbledon & Cannon Hill) or Morden Hall Park (NT but free) and a walk or bike over to Abbey Mills through Morden Hall Park is lovely. Further afield we go to Richmond Park or the Wetlands at Barnes, or out into the Surrey Hills. There's also a David Lloyd on the Bushy Road if you want tennis/gym/outdoor pool.
As I said before though, you'll be paying for the postcode in Buckleigh. My friend lives in a 4 bed, 2 bath end terrace across Martin Way and it's probably worth around 550-600k, the same in SW20 would be at least 100k more and the family friendliness is no less! If you look in Cranleigh or Tybenham roads (SW19) you're looking at £1m