Most of South Manchester is pretty diverse. Part of that diversity is provided by very transient populations, e.g. students, people who have just moved to Manchester and are likely to move on, overseas workers, etc. This is great and adds to the vibrancy of the area, but it can make longer term friendships more difficult.
There are also some large groups that tend to hang together - DD had lots of Muslim friends at primary school but has struggled to stay in contact as they are more restricted in what they are allowed to do and also much of their free time is very family / religion orientated. Friends that live in popular areas for Jewish families, e.g. Prestwich, have found the same - lovely families but tricky to be more than just pleasant neighbours.
I've just done a Rightmove search with your budget, these are good options in Withington for example:
www.rightmove.co.uk/property-for-sale/property-88326275.html
www.rightmove.co.uk/property-for-sale/property-88973045.html
The first property is near Old Moat and Cavendish primary schools - both good schools with diverse intake. Secondary options would be Parrs Wood / Chorlton High and possibly William Hulme. Both Parrs Wood and CHS are building schools nearby too.
Levenshulme has a lot of what you are looking for. I don't know much about the schools around there but some nice houses for your money:
www.rightmove.co.uk/property-for-sale/property-67922607.html
In terms of safety, kids play out around Withington and Burnage around the estates. Both Didsbury and Chorlton are in the top 10 postcodes for burglaries nationally though (covers Withington too). Like all cities, Manchester has muggings, burglaries and other crimes going on. The 'nice' areas have worse statistics as they are targeted for their likely assets and also are more likely to report crime. I've never felt unsafe around and about, but do have friends (all male I think) who have been mugged by groups of kids when out running / walking across a park. We do always keep all doors locked, make sure nothing is left out in the car but accept that there is a risk of crime.