你好吗？
How are you?

我很好。我很忙。

An adjective in Chinese is a predicate all by itself — it needs no 是. 我很好 is a complete sentence, and 我是好 is simply wrong. So why is 很 there? Because a bare 我忙 sounds like the first half of a comparison, as if someone else were not busy. 很 fills that gap and carries almost no meaning of its own: 我很忙 is the ordinary 'I'm busy', not 'I'm very busy'. To hear the real 'very', you have to stress it: 我很忙.

我很好，你呢？王老师呢？

呢 ne bounces the question you were just asked straight back, with no verb at all. Say 你呢？ after answering 你好吗？ and it means 'and you?'. Put any noun in front of it and it inherits whatever was last being asked: 王老师呢？ after a run of 忙 questions means 'and how about Teacher Wang — is she busy?'. 呢 is toneless and light, like 吗.

你忙吗？我不忙，我不累。

不 goes straight before the adjective, exactly as it goes before a verb: 不忙, 不累, 不好. Notice that 很 disappears in questions and in negatives — you ask 你忙吗？ and answer 我不忙, keeping 很 only for the plain positive answer 我很忙. And listen to 不 changing shape: 忙 has a rising tone, so it stays bù máng, but 累 falls, so it becomes bú lèi. Same story in 不错 búcuò, 'not bad'.

你好吗？我很好，谢谢！你呢？

Run the whole exchange aloud three times, changing the adjective each round: 我很好, then 我很累, then 我很忙. Always answer first and ask back second with 你呢？ Then say two negatives on their own — 我不累 and 我不忙 — and listen for the 不 changing: bú lèi falls twice, bù máng does not.