My info is 20 years old. Back then there was a corner shop and a chippy near the station and that was pretty much it. There were a few shops in Lee, also near the station.
Tons of stuff in Lewisham but be careful what you have on you when you shop there. Despite having the biggest police station in the Southeast, horses and helicopters, crime rates rocketed after it opened. Lee is a very nothingy place. Not much going on, a bit rough, not as bad as Lewisham.
Your best place for fancy shopping, boutiques, antiques and fancy food is Blackheath. Same for restaurants. I frequently got taxis there for nights out and they didn't cost much.
No, I don't think Greenwich Park is walkable from Hither Green.
It was leafy, suburban and pretty when I lived there, but also a bit of a no man's land.
The Victorian conversion flat I lived in backing onto the nature reserve was stunning.
Also, I remember the train service into London was pretty horrific with regular cancellations, look into that.
In terms of kids I know nothing, except to say there frequently seemed to be stuff going on in Blackheath on the heath or Greenwich. Greenwich also has a brilliant fireworks display.
If you need to be in that South East direction to get into central London, have you also considered Eltham? Some areas of it have a bad rep, but the older parts of it have stunning massive Georgian/Victorian houses and felt quite safe.
Please bear in mind this info is OLD, perhaps try to find some other threads on Mumsnet.