So once it is stable, there is no much point in fiddling with the tune ?
Yah, the whole point of autotune is that it picks an operating point that is stable, and that has a decent amount of margin around it (you don't want a slight change in weight and balance or aging of a motor to render it invalid. With the old autotune, there was a lot of uncertainty in the measurements and things needed a lot of tweaking... the averaging-based approach has mcuh less.
By moving the sliders you can erode that margin, but it's less likely to hold in all flight regimes. Turn down damping and you're more likely to get oscillation. Turn up noise sensitivity and you risk excessive coupling of noise into the output (and oscillation if you really turn it up). Either one-- trying to respond to noise with the motors or giving them oscillating commands-- makes motors hot and burns up windings and ESCs.
Regarding throttle settings what are our options: motor curve fit? throttle curve?
You could lower maximum throttle from the default 90%, sure.
I am trying to achieve 3minutes (!) of autonomy in my MiniOwl. It has already gone through a severediet : replaced every single screw with Titanium and Aluminium, lighter antenna, HQ props, there is not much left I can do except rewinding all 4 motors or go for lighter HV batteries (i.e lower C rate...)
Owl frames kinda suck. They have all the efficiency downsides from a restricted exhaust without the efficiency benefits that properly shaped intake can create. Micro quads making it for about 3.5 minutes of moderately aggressive flight is normal, sadly, and I'm not surprised a small owl type of thing does worse.