But she'd have done as much for theirs, any day.
She would have done as much for theirs (their letters), namely open them. The use of would talks about unreality. A queen's business is not to open her servants letters and in reality she would not have, but it means that the queen had a nice character and would have opened them if that had been part of her duties.
(they say she had a beautiful hand)
is open to interpretation
...She drew a real letter from her pocket, and held it out almost at arm’s length, like the old post-mistress in the village when she reads telegrams.
A native speaker also has to guess at what the author means by "her hand." If we are not familiar with the queen's handwriting...if we do not suspect 'hand' to refer to 'handwriting'...it is a natural assumption to think 'hand' refers to the queen's actual hand (the thing at the end of the arm). She could use her hand to excuse herself; and if she isn't wearing gloves or something else that hides her hand, her hand would be in plain view while reading the letter in the way 'the lady' demonstrates. Also, earlier it says "She held up her long jewelled hand."
But as two answers have said it refers to her handwriting, it seems this reading is not the only one, and perhaps not the most likely one. But when we read we often create meanings for ourselves which may not have been what the author intended. But this takes us to literary interpretation, which is beyond the scope of this site. Lesson: native speakers have to guess a lot too when reading older works!