In 1700s, why was 'books that never read' grammatical?
English in Defoe's time was different. Commas were used differently; relative pronouns were used differently; capital letters were used differently (although that last isn't evident here).
The that in the quote would be who today. It refers to I. A modern rendering would have to reorder the sentence in order to get the who nearer to I:
What should I — who never read half an hour in a year, I tell you — do with books?
In Defoe's time, it was preferred that the verb with its object was nearer its subject, and everything else was subordinate and came afterwards.
To answer your questions:
There is nothing special about the verb read and adverb never.
This question is moot, because the pronoun that doesn't refer to the books, but to the speaker.
Hopefully, explained here.
The comment and answer at time of posting are right about what the phrase means - it could also be written "What should I do with books? I've never read half an hour in a year, I tell you."
But I wouldn't overlook the Defoe factor. Daniel Defoe was an accomplished satirist, and in this extract he's writing in character as "The Gentleman". If the sentence structure appears odd, there's a good chance that this was - at least partly - the point.
This online edition has, on the front page (page 92 of the online document) the following :
"Usefull Observacions on the General Neglect of the Education of English Gentlemen with the Reasons and Remedies."
How can I interpret this curio so that it feels natural and intuitive to a reader in 2019?
Replacing "that" with "having" and introducing a pause helps me to read it naturally:
What should I do with books, having never read half an hour in a year, I tell you?