So you put him down and leave the room? Then you hear him awake for like an hour till he falls asleep? And does he fall asleep independently most times?
I would think that if he is just lying there when you put him down and not crying AND it takes him very long to fall asleep then he is UT and you might just need to add 10-15min to his A time. I refer to the two first naps that seems to be at home.
For the third nap I would try the sling. Do you have one? That will solve the running up and down, you could just do what you are doing and LO will fall asleep whenever he is ready and will sleep for as long as he needs.
It's hard for me to see from your EASY how long does he actually nap for. You count A from eyes open till eyes closed, including E.
I am still a bit confused as to when the A should start if he has a short nap and isn't due for the next E for an hour or so as there would start with the extra activity before the E.
Yes, that's true, A before the next A, then a little bit more A just to separate E from S and then you have your nap. This is normal. You start the A whenever he wakes up from his nap, even if it was a short one. You could of course try and extend the nap if you wanted to.