I'm just off Green Lane by Sidcup Road, heading towards Chislehurst. Green Lane itself is a bit busyish, particularly at rush hour, but the surrounding roads all seem nice. They are building a Premier Inn or such at Fiveways (Green Lane/Sidcup Road intersection) at the moment.
As for multicultural, the area looks predominantly white to me moved here from East London, but my next door neighbours are black, two down are Asian, a few doors further Asian, some Chinese... We own our house, either side of us are council houses. Not sure beyond that, although a few houses across from us have been for sale. At a guess, 50% owned/privately rented, 50% council. No idea on religions (although I think the NDN may be Christian...). People are friendly in my experience.
Lots of kids around. There's a small-children playground in Altash Gardens, and a larger-children playground up the hill from it, closer to Montbelle. There's a fab playground behind the library at New Eltham station.
I don't know much about Montbelle. The impression I've got is that it's not great, but all right. I think they've had a new head in the last few years, so there may be lots of recent changes. I don't know anyone who goes there - but my DC have been at the preschool at Mottingham Primary we lived opposite when DC1 started so most people I know with kids go there. I didn't get a chance to go to their open day.
We are around the corner from Greenacres Primary. I did go to their open day - it seemed like a really good school. Their stats aren't great because they have a Speech Impairment Unit, for seriously delayed students who come from all over the borough, and their scores are counted with the regular classes. The headteacher seemed good and I ended up leaving DC2 in their nursery while we toured - he didn't want to leave. 
Everyone local I've spoken to has sent their kids there and are enthusiastic - but I am .1 miles from it, so not surprising they all go there.
New Eltham station is good. They've put lifts in so both platforms are accessible with a buggy. Trains go to Cannon Street or Charing Cross, about every 15 minutes. Green Lane has the 160 and 162 more roundabout, which go Eltham High Street, New Eltham, Chislehurst and beyond and are very regular. The 314 (New Eltham station, down William Barefoot Drive) goes to Bromley, which has good shopping.
To commute to Blackheath, you would need to take any of those three buses to Eltham High Street, then either a train from Eltham or another bus. By bus it's about an hour. About 30-40 minutes with the train, depending on how the connection is. And Fiveways is incredibly slow during rush hour. It's usually much faster to walk to the other side of it, then get the bus.