Trouble with he first nap is he always falls asleep in the car it seems unavoidable.
Is there anyway you can avoid this? Do you have to be out and about at this time?
But he's been going to bed in his bed at bedtime and the past three nights has got increasingly less wakey (tonight I've not heard from him since 7.30!!!!!! And it's 9.30. Before I'd be in and out).
I'm glad he is able to settle in his cot now, that's great progress! I think working towards him going down awake in his cot like you have been and moving away from feeding/rocking to sleep is probably the way to go. Are you using ssh pat?
His naps are still quite short.
Are they? from the EASY above it looks like 3 1.5hr naps and a cat nap?
There is a 3 month growth spurt so this could be contributing to the disturbed nights too. Does he feed well during the day? is he taking full feeds?