Eat is an irregular lexical verb. It has the third person form eats /i:ts/, preterite "ate" (/eɪt/ but often reduced to /et/); the past participle "eaten" /i:tən/ and the -ing form "eating" /i:tɪŋ(g)/.
Phrasal verbs we have a list. e.g. eat out.
"ate" is a homophone with the number eight.
If Chinese students ask "have you eaten?" or "did you eat?" they are not really asking about your lunch but translating from Chinese "你吃饭了吗?" (trad: 你吃飯了嗎？ Pinyin: "nǐ chī fàn le ma?" - literally "you eat rice [past particle] [question particle]?), and what they really mean is "How are you?".