I know you've asked this question (or questions similar to it) several times now, and I've been waiting to answer in hopes that I might be proven wrong. But there's no evidence that I'm aware of for European or Muslim troops adopting each other's equipment during the Third Crusade; one expects that, if they were going to, they would have done so some time in the previous one hundred years, rather than during an active campaign. The only source I'm aware of that even describes individual equipment to any great extent is Baha al-Din, who emphasized the protective qualities of the Europeans' armor, consisting of both mail and some variety of felt vest that excelled at stopping arrows. I can't emphasize enough that most medieval sources are terrible for understanding the small details of military life. They are, in the main, concerned with narrative (and glorification of aristocratic sponsors). Tables of equipment, casualty reports, payrolls: we have none of these things for the Third Crusade.

Probably not, no.

Equipment in crusade warfare was based on the user's tactical approach to battle; these were fairly incompatible between the Christian and Muslim armies.

It is likely that Arab use of longer lances stemmed from their experiences fighting Franks: the equipment of the Crusader forces favoured close combat and so lances allowed for Arab fighters to engage from a greater distance. However, these were not Frankish lances, nor were they copies. These lances, especially those employed by the Turks, were much lighter.

Likewise, the Muslim armies were equipped with shields but these were not the shields of the Franks. Arab shields were smaller (round, not 'kite-shaped') and also lighter. The theme of weight reduction should be becoming self-evident here: the tactical approach of the Muslim armies was overwhelming that of mobility.

The Frankish milites were increasingly armoured. Mail flowed increasingly over the extremities, into considerable skirts, over hands and into loops around the neck and face. A Muslim outfit not serve a Frank, nor would a Frank's equipment benefit an Muslim, because each is tailored very specifically to their respective culture's ideas on warfare.

Would weapons have been re-purposed on the battlefield? Logically we would yes, almost certainly. Do we have evidence that it was widespread or institutional? No.

For more information please consider Crusading Warfare, 1097-1193 by R.C. Smail.