Hi there
What happens in the morning on the way back from school after he falls asleep in the car? Does he wake up himself at 9.30 because you arrive home or are you waking him?
On that second nap have you tried a W2S (wake to sleep) to sooth him through the transition between sleep cycles at around 30-45 min?
have you tried a slightly longer A time before that second nap? Maybe 3hr 15 rather than 3hr?
For he nights, I'd first suggest you try and have a think about what may be waking him. Too hot? Too cold? Noises?
It may also be that his sleep in general is disturbed but we can end up very focused on the baby rather than the bigger picture so just have a think about it see if anything occurs to you.