1. Did you finish reading that, Harry Potter novel?

Question

(What is the part of speech of 'that' in the sentence? Do we need "comma" after 'that'? Is the sentence correct?))

Answer 1

No comma needed.

The word "that" is usually a demonstrative pronoun.
http://grammar.ccc.commnet.edu/grammar/pronouns1.htm#demonstrative

In this sentence, it's serving as an adjective (see the second section and examples in the link above) modifying "Harry Potter novel"