1) A photographer uses a telephoto lens for his camera when taking photos at a long distance.
You could also use 'lengthy' distance, or simply 'at a distance'. (I've never seen someone use big and distance together in a sentence.) Although technically it may not be wrong, it is not a typical usage. It is somewhat like stating, "I made a big distance telephone call," rather than, "I made a long distance telephone call."
2) "However, by this date he had become one of the three leading members of a company of actors called 'The Lord Chamberlain’s Men'.
It is a name as a whole. Now had the 'the' been within the name instead of the front, it wouldn't be capitalized.
3) "This company was under the protection of Lord Chamberlain, a powerful nobleman and an official at the Queen’s court."
You are refering to an individual, not the tittle of the individual.
4) Should there be an article before "legend" here?
"According to legend he was apprenticed to a butcher."
No, if you were refering to the "Legend of Sleepy Hollow" or some specific, named legend, then you would use 'the'.
You might also need it if should you say, "According to the local legend, he was apprenticed to a butcher."
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