"Die Frau isst mit ihrem Koch."
Translation:The woman eats with her cook.
"ihrem" cannot be translated as "your", because the formal form is capitalized - "Ihrem".
OK so tell me this - what happens in SPOKEN German? No capitalization when you're talking!
You just have to work it out from context. Generally it is easier to work things out in a conversation than in these out-of-context examples.
Usually you can figure it out through context, but I'm sure that sometimes someone may ask for clarification. For example, someone may ask, 'Meinen Sie, meinen Koch oder der Koch der Frau?' (Do you mean my cook or the woman's cook? I'm assuming I got the German right) Kind of like in English, when someone uses the word 'you,' sometimes you need to ask the speaker, "do you mean me specifically or just people in general?"
so the third answer here: with YOUR cook is -- WRONG. The choices for you here (in the dative cases) being deinem, Ihrem (always capatalised?) and eurem. Is this correct?
Nope, 'their' in dative is also 'ihrem'. 'Eurem' is 'your', but for the plural 2nd person. So you would say "Die Frau isst mit eurem Koch" if talking to more than one person, all who have the same cook.