Learn Russian: Вы отвечаете правильно и ясно.

In Russian, you can turn many adjectives into adverbs of manner (words that answer the question как? or "how?") by replacing the adjective ending with -о. So, the adjective for "correct" (правильный) becomes the adverb "correctly" (правильно).

How can I tell that правильно is an adverb and not a neuter adjective, since they both end in -о?

You look at what the word is describing! Neuter adjectives describe neuter nouns (like "a clear morning"), but adverbs describe actions (verbs). Because правильно and ясно are describing the verb отвечаете (how you are answering), they are adverbs.

Do adverbs always go after the verb like they do in English?

Not always! Russian word order is very flexible. You could easily put the adverbs before the verb and say Вы правильно и ясно отвечаете. Putting them at the end of the sentence, like we do here, just puts a little more emphasis on the result of how you answer.

Why does the verb отвечаете end in -ете?