On Duolingo you will find that Lei, Suo, (Suoi) are capitalized to indicate formal you. Lowercase is simply he/she/it. Such capitalization is far more frequent in official documents than in everyday Italian. However, Accademia della Crusca and teachers of Italian as a foreign language support this distinction, and we have implemented it for pedagogical reasons. Imagine if today was the first day you decided to learn a second language, how confusing it would be to deal with lei (she) and Lei (you)!
They are the same, in context it would be clear whether you were talking about a male or female. It is possible to make a distinction by saying l'amore di lei, her love, or casa di lui, his house, but these are particularly emphatic. You're perfectly safe sticking to suo in most cases.
Is there a separate present continuous in Italian? The system wouldn't accept "His horses are drinking the water". There are inconsistencies elsewhere such as when I translated "pesce" wrongly, I was told to use "seafood", yet in a later lesson it told me that "seafood" was wrong and insisted on "fish".
The gender of the possessive does not refer to who owns the thing. The gender of the possessive refers to the thing, just like any other adjective.
Since "cavallo" is masculine, it takes masculine adjectives, including articles and possessives. It is always:
il mio cavallo
il tuo cavallo
il suo cavallo
il nostro cavallo
il vostro cavallo
il loro cavallo
No matter what the gender of the owner is. Therefore "suo/suoi/sua/sue" can be "his" or "hers".