So, I ran the bike everyday to go to work (a 10 miles job ). So when I arrive at work the tranny is a bit warmer (it's cold here...). The gears are much much better with the hot gearbox. Going down is quite perfect except sometimes from 3 to 2. I ran the bike for about 35 miles and the box does not get hot nor noisy so I think end play setting is good. I checked the oil level often, and no leak either.
As i installed a reinforced German made spring on the pawl, I contacted the seller and explained my problem. He told me that this is caused by the thicker spring. The wire used has a bigger diameter than the stock part. So the two parts rub more than they should and the spring is not strong enough to get the pawl back. (this explain why it's better when the gearbox is hot, play is more important and oil not so thick so does not stick as much).
He told me I can use the gearbox as it is and save the installation of a stock spring when I have to open the box again.
As the garage is freezing cold and the wife complains on the long hours spent in it instead of shopping or going to places together, I'll run the bike as is and wait 'till spring or another mechanical problem to fix it proper.
Thanks all of you for your help and sound advices !