Is it possible to install a thru axle on a bike, which is a few mm to shorter than the bike frame's thread? I installed a thule thru axle on my frame, but it is a bit shorter than the original one (newman, see pic).
I think 4-5mm to short.
Furthermore, the original one is CONIC, and the replacing thule one is 'flat'.
Can this hurt my frame?
You can never replace a conical interface thru axle with a flat shouldered one. That's just the wrong part.
Lack of contact area may cause the axle to loosen or the frame to become damaged. But what will probably be a more immediate problem is that most frames with a conical interface have a hole for the axle shaft that's a pretty loose fit, and often gappy by a millimeter or more. The reason is they're using the conical interface to locate the axle and prevent it from squirming under load, which is actually a much better design than relying on the tolerances of a plain hole, and probably the only reason it's not universal is because of cost. Don't sabotage your bike by plugging in the wrong thing.
Some aftermarket thru axles are sold with a conical washer and two lengths are given. One is for with the washer, one without. It's a way of halving SKUs needed.
As for your original question of missing thread engagement, it depends more on what engagement length remains, not what's missing. That said, wheel retention is probably the literal worst thing on a bike to play thread engagement safety guessing games with. I doubt anyone who knows what they're talking about would be comfortable stating a generic minimum safe thread engagement number, other than 100%. In practice I've been allowing no more than about 0.5mm of missed engagement and have mostly been erring long.