In the following sentences, do we need " , and " to clarify that both verbs are for one subject?
1- " Alice went to the restaurant, and ate lunch".
2- "Alice went to the restaurant, ate lunch, and got back home"
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In English, two verbs that refer to the same grammatical subject are normally connected with "and", and the two example sentences in your question are correct.
(I consider these sentences as examples of American English. In British English, there is usually no comma before "and", but see also Commas, subject and verb on English Language & Usage SE.)
There are languages that use "and" less frequently (e.g. 和 in Standard Chinese), but if you left out "and" in the first sentence, a native speaker of English would normally feel that the sentence is incomplete until you add "and [x]".