Given a particular Twitter profile, that contains tens of thousands of tweets, what is the best approach for:
a) downloading the text (with date) of all tweets
b) search for a specific word in all tweets
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If you want to search for a specific word (which, I believe, is what you meant by 'work') you can use Twitter's Advanced Search. I've attached a screenshot which includes fields both for Words and for People where you can specify the word(s) and the account(s) you want to search through.
You can also see the related question: How do I search within someone's tweets?