Brentford is in the Borough of Hounslow, as is most of Chiswick. So schools come under the same applications process.
Brentford, imho, is less expensive than some areas mentioned because it is further out of the centre of London, is more recently 'up and coming' (through development and investment in the area), has no tube station, just a mainline train station.
You are right that children living in one borough would usually not go to a shool in another borough. In general terms, children go to the nearest school in the borough in which they are resident.
In reality (particularly around here), many parents are aware of which are the best state schools in the local area and try everything they can to get their children into a particular school, even if it is in a different borough. Some parents will rent close to a good school in order to get a place. I know others who remain on the waiting list and get their older sibling in on an occasional place that comes up, this then bumps up their younger child on the waiting list.
Places at any of the outstanding schools here are very oversubscribed - even more so with the current financial climate and parents struggling to pay private school fees. House prices near outstanding schools are also higher.
The admissions criteria for offering places in Reception in Ealing. These vary according to the school you apply for and depend on what type of state school i.e. a church school will have certain religious requirements like baptism and regular church attendance, etc.
Entry to private schools will be dependant on the date on which you registered i.e. 'first come first served', ballot or by assessment. Many will register with a private school before their children are a year old.
With regards to a car, there are lots of car clubs starting up now. Also Treetcar. Basically, you pay a joining fee, then just book a car for an hour, half a day, weekend, etc and pick your car up from it's parking space, then return it when finished.