1. Two rabbits and a gerbil (lives, live) in big cages in our classroom.

2.Either my brother or my sister (is, are) waiting for me.
3.Neither a plum nor a pear (is, are) a citrus fruit.
4.Either an orange or a grapefruit (is, are) good for dessert.

1.lives
2.is
3.is
4.is

Can you please tell me if these are correct?

Those are correct.

Thank you very much.

You're very welcome.

If I have a sentence with plural subject and singular subject joined by nor does the verb agree with the subject nearer the verb? The sentence starts with neither.

Neither the harsh winds nor rain (affect, affects) this tree.

affects

Winds is plural but is rain considered plural also?

The verb is always singular with the "either-or" construction.

Your verb, "affects," is correct because it's singular.

http://grammar.ccc.commnet.edu/grammar/sv_agr.htm

Read #5.

Neither my cousins nor my aunt (wants, want) the salad.

So would the answer to this one be wants?

Right.

Yes, your answers are correct.

When determining whether to use "is" or "are" in these sentences, you need to consider the subject or subjects of the sentence.

In sentence 1, the subject is "Two rabbits and a gerbil," which is plural. However, when referring to multiple animals as a group, we use the singular verb form. So, the correct answer is "lives."

In sentence 2, the subject is "Either my brother or my sister," which is singular. Therefore, we use the singular verb form "is."

In sentence 3, the subject is "Neither a plum nor a pear." Again, this is singular, so we use "is."

In sentence 4, the subject is "Either an orange or a grapefruit." Once again, this is singular, so "is" is the correct verb choice.