I realised I missed the rest of your questions when I saw nottinghillbill's post (we also moved from West London - Shepherds Bush - me and Ealing - my wife).
We also love it here. The areas is really good for young families (I'm not so sure about secondary schools as have not researched that extensively).
Great Parks close by, often with great views: Blythe Hill, One Tree Hill, Dulwich Park (boat lake and all), Crystal Palace Park (Dinousars, small farm and all), Ladywell Fields (has a stream running through it) and many more. Kelsey Park in Beckenham is further afield but saw a Kingfisher there, and there are also menat to be some on the River Pool I think.
Horniman Museum must be on of the best kids museums, and interesting for adults also, they are brilliant. Their gardens are also really nice and they have a small farm now. They always have different activities on for children, it truly is an amazing museum for children.
Getting into town (whilst getting busier) is really easy (10 minutes to Canada Water for Jubilee line \ head straight to Shoreditch etc). Trains to London Bridge I'm not sure on with Southern but you can get there and then hop to Charing Cross.
Forest Hill pools local swimming pool, but no far also Lewisham Glass Mill, Depford is really good for family swim on a weekend.
Safety wise, as with everywhere in London, nowhere is 100% safe, but I personally feel it is much safer here than W12 where I used to live.
We've discussed schools of course, and the bad thing is the small catchment areas. If you get in however that is a blessing, I went to school where people came from all over London, whereas here most of my sons friends live walking distance or a very short drive from our house - great for playdates and getting a nice community feel (if you do move here you will be amazed by Halloween).
There is a really good community feel, the local library has just been saved in Forest Hill, have a look at the Forest Hill Society and all they have done (Michael from there posts on here).
There are lots of great child friendly restaurants - Babur (buffet on Sunday free for under 8s), Querce on Brockley Rise they love kids, Bona pizza in FH and many more. East Dulwich is just a short ride away.
If you do move here, my last tip, if you want your child to do gymnastics, is put them on the waiting for Ladywell Gymnastics in Bellingham, an amazing club but waiting list closed for 4-7 years olds, but if you get in at a younger age you will be moved up.
There are lots of after school clubs, lots of pre-school at hoc groups to meet people so I'm sure you could meet and make friends quickly.
You can also drive to the coast somewhere like Camber Sands relatively quickly for a day out or get the train easily to Brighton.
I've forgotten loads of things but if you move I think the chances are you will love it - everyone is different of course but most people who have moved here seem happy.
Good luck