Did gamboised cuisses appear in the 12 century ? this new protection allow the shield and the hauberk to be shorter ? Did this enhance the mobility of the soldier ?

Gamboised cuisses? Do you mean these, Alexis?
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The current evidence seems to indicate it was a bit later, in the first or second decade of the 13th century. The Maciejowski Bible shows an infantryman putting on a gamboissed cuisse, even though he doesn't have mail chausses, and likely isn't usually mounted. Like gambesons, they might have been the sole leg protection for burghers, and an additional defense coupled with mail for mounted men.

Leg armor, including mail chausses, aren't very common until the second half of the 12th century. The adoption of mail chausses and gamboissed cuisses worn together might have meant a longer shield was no longer necessary.

Ok, nice, So gamboised cuisses need to be thick enought to stop a one handed sword strike, did anyone try it ? :p

Another question about gamboised cuisses : It seems only the thigh and knee was protected by more than mail and underpadding, so can we say that lower legs was not a usual target for bludgening strike and projectile and spears was stopped by mail and underpadding alone ?

There is no evidence for them being laced up the back either. The Maciejowski Bible example appears to be a sewn tube (conical bore) pulled up the leg.

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Mobility and costs are important factors. The English don't require burghers and free men to have gambesons until the 1181 Assize. The long skirts offer some protection to the thigh, but not to the inner thigh where the femoral artery is closer to the surface. A cut making it beneath the gambeson could be deadly within a few minutes.

The cost of maintaining arms which might be rarely used was a burden on the non-noble classes. Marching for several hours per day wearing a few additional pounds or kilograms of mail on the legs would be a burden for those on foot, so armor was transported by wagon. Mounted men were more prone to being hit in the legs than foot soldiers, but could manage the extra weight since they weren't walking. The risk of a broken shin, even with mail, was noticable enough for shynbalds to be worn over the front of the tibia by 1240-1250, though a broken shin might not have as much impact on a mounted man, who could still flee, as it would on a foot soldier.

Thigh armor seems to have improved enough by the first decade of the 14th century to sometimes replace the mail chausses with mail chaussons, from hose to socks.

Would the padding over mail reduce wear an tear of a saddle as opposed to just plain mail? I don't think that was the primary consideration that led to them being used but it sounds like a possible added benefit.

I think all of the "stretch" that's needed comes from the compression of the loose cotton fill. You could always cut the fabric on the bias for a bit of give, though the King's mirror specifically calls for them to be made of stout linen (lérepti) in the same manner as the gambeson. Thom Richardson cites 14th century records for gamboissed cuisses covered in patterned silk or embroidered examples. I've never seen a reference for using wool for them.

En þenna um búnað þarf maðrinn sjálf að hafa: góðar hosur ok
linar, görvar af blautu lérepti ok vel svörtuðu, ok taki þær alt til
brókabeltis en utan yfir þær góðar brynhosur svá hávar at maðr
megi gyrða þær um sik tviföldum sveip; en utan yfir þat þá þarf
hann at hafa góðar brynbrœkr görvar með lérepti at þeim hætti sem
fyrr hefi ek sagt; en þar um utan þarf hann at hafa góðar knébjargir,
görvar með bykku járni ok með stálhörðum nöddum.

But this grim equipment is required of the man for himself: good hose and
loose, made of soft canvas and well blacked, and take them up to
the trousers belt but outside of them good mail hose so high that a man
may gird them for himself double wrapped; but outside of it he then needs
to have good armored breeches made with canvas, in the manner
which I have said before, and on the outside he needs good knee-plates,
made with proven iron and steel hard nodes.