Translation:They do not like peanuts, so they do not eat.
In English, this sentence technically correct, but it feels unnatural. I agree that it should end by specifying what they don't eat: "the meal", "anything", or "at all". and I also agree that the last phrase could be in past tense to be more natural. These sentences would all be very natural, in my opinion:
"They don't like peanuts, so they didn't eat anything."
"They don't like peanuts, so they didn't eat the meal."
"They don't like peanuts, so they didn't eat at all."
"They don't like peanuts, so they didn't eat anything at all."
Not necessarily; depending on context, peanuts might be an ingredient in a meal or snack in a certain situation (as in food with satay sauce).
Eat is a transitive verb in this case and demands an object. The best answer would be "...do not eat any."
so they don't eat (anything/at all - don't eat any/them would imply peanuts)
“Them” is necessary at the end of the sentence to clarify whether the people do not eat ‘the peanuts’ (them), or do not eat at all (in which case “at all” should be added so that the sentence is not left incomplete).