The horse referred to in the Hadith in described in the Arabic text as 'Mudhammar' i.e., a specially trained and well-looked after horse. Such a horse is first fattened by supplying it with large quantities of food. Then slowly the quantity of food given to it is decreased and it is made to sweat in a hot room until it loses weight and thus, becomes fit for running.
The Arabic word "dhil" translated here as 'shade,' in fact, refers to the expansiveness and vastness of the tree. It does not necessarily mean shade in the ordinary sense, as there will be no sun or heat there. Here the expansiveness and vastness of the trees of Jannah has been mentioned. Allah can do what He likes and create whatever He likes.