Duo's course emphasizes that "yiyor" is a present progressive thing, something that's happening right now. So Duo wants you to say "Senin tavuğun peyniri yiyor" means "Your chicken is eating the cheese."
Aorist "yer" actually COULD cover "would eat." It could also cover "eats." It's all contextual. It's called the "wide tense" (geniş zaman) in Turkish because it actually spans over such a range of meaning.:)
(Unnecessary but interesting detail for aorist/"wide tense": http://www.turkishlanguage.co.uk/aoristpos.htm )
If we wanted to say "You eat the cheese", you're correct that we'd use yersin: "(Sen) peyniri yersin."
However, in this case, "you/sen" is not the subject doing the eating. The subject is actually "your chicken": "Senin tavuğun". Because "your chicken" is a he/she/it ("o" in Turkish) we use the third-person form of the verb: "yer"
Hope that helps :-)
"Senin" is casual singular-- so you're talking to someone around your age that you know well, or to a child (whether you know them or not).
"Sizin" is either formal singular or plural (both casual and formal). So you use this when talking to any group, or to a stranger, boss, teacher, elder, etc. When in doubt, it's better to start with a respectful "siz" until they tell you that you can be more casual. :-)
If there is nothing in the sentence to suggest who we're talking to, Duo will accept either one as long as the whole sentence matches: