Could be a masculine noun such as τους κουραμπιέδες (a kind of baked item often eaten around Christmas).
"κουραμπιέδες" reminded me of the turkish word "kurabiye" (a sort of biscuit) and indeed it seems to be of turkish/arabic origine.
No; these short forms such as με σε τον τους etc. are only used before a verb, or right after an imperative (command) form.
If you want to put the pronoun after the verb, you'll need the full form: Τρώω αυτούς.
Well, you can if you're speaking Greek in Cyprus (a fact usually ignored); but anyway this course is about Standard Greek so you should concentrate on that.
Indication of the pronouns: