“Very often indeed they would first give us a song,” versus the modern text: “First they would often sing to us,” an example of syntax, diction, or idiom?
An idiom is a group of words whose meaning cannot be comprehended from the definitions of the individual words. “Very often indeed they would first give us a song,” and “First they would often sing to us,” both seem to literally say " The first thing mentioned people would do is sing to the narrator." We can gather the meaning through the individual words, therefore the sentences are not examples of syntax.
Diction is the style of speaking a character uses, typically falling under the categories of formal and informal.
Formal diction is used in speeches, essays or reports, and typically involves intelligent use of words with the correct grammar and tone required. Informal diction is used in everyday conversations or messages, usually between friends, and typically involves slang and getting your message across quicker compared to following the normal language rules.
An example of informal diction would be:
"The guy has his leg done for, he probs needs surgery, ASAP."
The opposite would be formal diction, which would look like this:
"The patient has experienced acute trauma to the right femur, and must have surgery stat."
The sentences in question don't show a remarkable change in word choice and style, remaning quite formal in both instances, therefore the quotes could not be examples of diction.
Syntax is "the arrangement of words and phrases to create well-formed sentences in a language." It typically involves the ordering of the subject, object and verb in a sentence.
Looking at the first sentence of the question, “Very often indeed they would first give us a song,”, we can see how the order of the words is much different from "First they would often sing to us." Because the order and arrangement have changed, I mark this down as an example of syntax.
I hope I helped!