@Scagnetti Thanks for the input. We've been working on the crate since day one. We actually do crate games and impulse control drills daily, with and without the crate. Things like hand feeding with the food right under his nose, but he is not allowed to touch it; or dropping things around and moving through them; or simple things like waitting for my command to start eating, sometimes more than a minute (he just stares at me waitting for the release); or placing food close to him and just leaving the room (again, he just sits and waits). Also simple sits & waits before doing anything or when he is agitated (going out, getting back in, getting in the car, playing with toys, getting inside a room he wants or basically everytime HE wants something)
Now of course, we are talking about tens of seconds (under a minute or a little over), but what what can you expect from a puppy. He is doing great in this regard.
He also sleeps fine in the crate both during the day (we don't always have him lay in the crate, but for exercise purposes we have him sleep there once) and at night (with us in a separate room). If I have something to do and ask him to get in and stay there for a little while, he does. A few days ago he went inside alone and fell asleep for the first time ever, I was amazed. Didn't happen again since, but still.
I've noticed that he follows us around way too much. I fugred he was just curious, but it may be more than that. He jumps out of bed anytime both of us leave the room, or wakes up when one of us leaves. Since yesterday, we started to show him back where he was and ask him to stay (it takes a few tries before he does), so we are starting with that, alongside with short leavings (today we had in the crate with a kong for about 30 minutes and we just did what @elbrant suggested, aka leave for 1-3 minutes and come back.. after 2-3 times he didn't even bother checking on us). We'll continue this way, trying to increase the duration.