How will one know when to differentiate between this and that as the translation for das?
If it's by itself, you can only tell from context -- Duolingo should accept both translations.
You can be more precise by using das hier to mean "this" and das da to mean "that", e.g. Ist das da dein Haus? "Is that your house?"
Or together with a noun, you'd have e.g. das Buch hier "this book" versus das Buch da "that book" if you wanted to be more precise than just das Buch "this/that/the book".
Use dein, deine, deinen when speaking about something that belongs to the one person you are speaking to.
Use euer, eure, euren when speaking about something that belongs to the more-than-one people that you are speaking to.
It's a bit like the difference between "my" and "our" or between "his" and "their" -- whether the owner is singular or plural.
As for the ending: no ending for masculine nominative, neuter nominative, neuter accusative; -e for feminine nominative, feminine accusative, plural nominative, plural accusative; -en for masculine accusative.
For example, if you are speaking to one person, you might say Ich sehe deinen Hund und deine Katze und dein Pferd. Deine Tiere sind alle klein.
While if you are speaking to several people who own those animals together, you would say Ich sehe euren Hund und eure Katze und euer Pferd. Eure Tiere sind alle klein.