you can travel with less than 6months left on your passport - in theory you can go through 'borderless' Europe without needing a passport, although everyone, especially airlines, ask for one as proof of ID.
i'm sure the info posted here on the US is correct, and less than 3 months is still OK.
I check + send passport forms at work (post office). Your son won't automatically be called for an interview - at the moment the numbers called in are very low, and usually only if you've held no other passport before. Your son has had a British passport (as a child) and the ID checks would have already been done when that was issued.
If you apply through your local Post Office on Monday, you should have a passport back in 10 days. We can't guarantee it, but it's the normal turn-around time.
It costs �7 to process this way, but I'm sure it will be worth it.