Translation:The boy is eating a lemon and a lime.
No, saying "a lemon and lime" does have a slightly different meaning. It either implies that the lemon and lime are one (like in a soda) or that the lime is something that using "a" with would be grammatically awkward, such as when it is used to mean limestone. "A lemon and a lime" establishes that the two are different entities.
eet is for the singular, eten is plural:
ik eet / wij eten jij eet / jullie eten hij/zij/het eet / zij eten
think of it like eet is eats and eten is eat. i, you, he/she/it eats. we, you (all), they eat.
therefore, hij eten is saying (singular) he (plural) eat and is grammatically incorrect. should be hij eet.
hope this helps :)