Do I need to sign the words 'did' and 'you'?
No, you don't need to sign 'did'. In ASL, yes/no questions are indicated by raising your eyebrows. While you can point to the person for 'you', simply signing 'eat' while looking at them with raised eyebrows is usually enough to ask 'did you eat?'.
Is this the same sign as 'food'?
Yes, 'eat' and 'food' use the same handshape and location. Generally, 'eat' uses a single, larger motion, while 'food' uses a smaller, double tap. However, when asking 'did you eat?', the context and your questioning facial expression make your meaning perfectly clear.
Why do I have to raise my eyebrows?
In ASL, facial expressions act as grammar. Raised eyebrows and a slight forward head tilt are the specific grammatical markers used to ask a yes/no question. Without this expression, you are just stating the word 'eat' rather than asking a question.