Grammatically Correct Sentence: Adapting Quickly to New Environments

Yes, the sentence 'Having lived in Taipei, Shanghai, Shenzhen, Boston and New York, I adapt quickly, have high levels of curiosity and am able to cross culture with ease.' is grammatically correct.

This sentence effectively demonstrates your adaptability and cross-cultural skills by using strong verbs and specific examples. It also highlights your curiosity, which is a valuable asset in any environment.