Why did the pronoun "هذا" (this, masculine) come instead of "هذه" (this, feminine) in Allah's saying in the Hadith Qudsi: "This is between Me and My servant," even though the reference is to "prayer," which is feminine? And why was it followed by: "and for My servant what he asked for," even though the servant did not ask for anything?

There is no problem in expressing the verse with the word "this," because what is meant is the masculine utterance, or the humility and submission that the utterance conveys. Al-Qurtubi mentioned that "this" refers to the servant's humility before Allah and his seeking assistance from Him, which implies the glorification of Allah. Abu Dawud's wording in the Hadith, "And this is for My servant," confirms this meaning. As for "And My servant shall have what he asked for," it refers to what the servant supplicates for thereafter, seeking guidance.