Learn Arabic (Modern Standard): كيف تنطق هذا الحرف؟

In Arabic, when you want to say "this [noun]", the noun following a demonstrative word like هذا (this) must be definite. If you left off the الـ, the phrase هذا حرف would become a complete equation meaning "This is a letter", rather than the phrase "this letter".

What is the root of the verb تنطق and how do I change it if I am asking a woman?

The root is ن - ط - ق (n-t-q), which relates to speech and articulation. It is a standard Form I verb. The word تنطق (tantiq) is conjugated for "you" (masculine singular). If you are asking a woman, you add the feminine suffix ـين to the end, making it تنطقين (tantiqeen).

What vowel ending (haraka) would الحرف take if I read this sentence aloud with full formal grammar?

It would take a fatha (an "a" sound) at the end, becoming الحرفَ (al-harfa). This is because the phrase "this letter" acts as the direct object of the verb "pronounce". In Arabic grammar, a definite noun directly following a demonstrative pronoun like هذا copies its grammatical case, which here is the accusative case used for direct objects.