Yes, you are correct. The second one is more appropriate in contemporary English. The first one requires a comma after the word 'work'.

If you like that one better, use that one. Both are traditional ways to say it. Not quite the same. The first one is older, probably less common now. .

You can compare the two sets of sentences below to see why "it" can appear in the relative clause. 1. [There are People] + [It is their work to sweep the street] => There are people whose work it is to sweep the street. 2. [There are People] + [Their work is to sweep the street] => There are people whose work is to sweep the street. The second sentence of (1) has the placeholder "it" which gets carried over to the relative clause, while (2) doesn't have "it" in the original. The second sentence of (1) and (2) have the exact same meaning, but having "It is" in the beginning makes the sentence read smoother. Another example: [He is the caretaker] + [It is his job to watch over the facility] => He is the caretaker whose job it is to watch over the facility.

Either is fine. The structure using a "dummy subject" - in this case "it" - to allow important words to be pushed to the end of the sentence is widespread and sometimes useful in English. https://dictionary.cambridge.org/grammar/british-grammar/about-words-clauses-and-sentences/dummy-subjects. http://users.utu.fi/micnel/endweight.htm