Hi there Worldita, welcome to the wonderful Benelux mumsnetters.
Do please read the long post I wrote on 1st March listing the communes where there are more schools with a good mix of expats. It's not just Etterbeek which has schools with a decent mix of expats, the further east and south-east you go, the more schools with this mix you'll find and also more places too, due to the turnover of expats and also the fact the east isn't suffering from overpopulation / lack of school places like other areas of Brussels region and Vlaams Brabant.
The whole of Brussels region has good public transport, but it depends on where you want to get to and what speed. The communes connected by metro have the fastest connections. There are a few tram lines with higher frequencies / dedicated lines on part of their routes : 3,4,7,25,39,44 are faster than other tram routes. Hope I haven't missed any.
Where is work, as that might have a major bearing on where you choose to live? Down to exact address would help with suggestions, plus the maximum commuting time you'd like.
If you want to see how green a commune is, 2 ways of looking. 1) wikipedia entry usually lists population density. 2) google maps, type in commune name and google will show you the limits of the commune and you can see how much of that map is green. As you can see, Etterbeek is densely populated and very little green, although on the edge of the Cinquantenaire park.
You can live without a car in Brussels region, but much is down to attitude. I don't own a car and never have done. I get use of a car approximately 3 days a month. I don't miss not having a car at all in the region. We live next to the tram line and near to all the children's activities and school for the good reason, those over 10 years old can be very independent and we don't spend our afternoons in traffic jams and don't spend much time travelling at all. I miss the car when I need to get the children to far away activities where the public transport can take 3 hours instead of 1 hour in a car. There is a car rental system called CAMBIO, I'm too coward to use it as I've never driven with a left hand wheel, well almost never. But if you think you might need a car occasionally, consider CAMBIO.
Shopping delivery, well not much of that, I use the tram and bag on wheels or a bike and bike trailer.
When looking for schools, do consider all types and there are 4 of them, unless you really want to make your school search hard. Very few Catholic schools are very Catholic at all, the Jewish and Muslim schools indeed would be hard to attend if you weren't of that faith, but these make up only 5 out of around 300 schools in Brussels region at maternelle / primaire level. So look at Communal, French Community controlled, Catholic/religious controlled, non religious controlled (only 8 of them with extra fees of 1-1.5k per year).
If your children are likely to be heading to EEB, then don't hold on them getting into any other school if English other than Laeken. The travel times there from areas like Chant d'Oiseau area of WSP are 30 minutes ish, so not as bad as expected.
Properties start coming onto immoweb around 3 months or less to being available. Do NOT rent until you're settled on a single area and 1 or 2 school places in that area. If you're unsure, then choose some areas where there are 1 or 2 school places AND also enough rental properties you are likely to find somewhere to live near school closer to the time you move. We had a year to plan our move. In the Autumn I visited the areas I'd chosen and narrowed it down to a 1.5km radius and schools in that radius. I did visits to some of them, immediately ruling some in and out (in the days you didn't need to enrol almost a year in advance like now for many schools). In the March, we returned and enrolled in a single school, after 2nd visits. By April, I'd spotted an advert for a house 300m from school and took it, signing 3 months in advance of moving. I would not advise doing this unless you're 99% certain of the school and house and have a plan B if school goes horribly wrong.