Is ‘you’ correct or ‘your’ in the following sentence? Why? TIA

Both are correct, but “I appreciate your stopping by” is much more formal Same with the 2nd one. Both are correct, but “If you don’t mind my asking” is very formal / old-fashioned in a way. It’s called a gerund — when a verb acts as a noun 

Historically in a number of germanic languages it was rude (or at least over-familiar to a stranger) to address someone directly as 'you', so there are all sorts of linguistic ways around this. My guess is that referring to 'your stopping by' is an example of this, but I'm happy to be corrected if somebody knows better.

I have several notes here: 1) In strictest grammatical terms, one might say that only the possessive "your" or "my" is correct, as this makes "your stopping by" or "my asking" into a noun phrase, and we need a noun phrase since we would like it to be the direct object of the main verb ("to appreciate" or "to mind"). 2) Functionally, VERY few native speakers are even aware that 1) is a thing at all. I wouldn't be surprised if you said "I appreciate your stopping by" and a well-meaning friend gently "corrected" you by offering, "I think you mean <*you* stopping by>." 3) This article from the LA Times/Burbank Reader gives a thorough treatment of the topic: [https://www.latimes.com/socal/burbank-leader/opinion/story/2019-12-05/a-word-please-fused-participles-can-suggest-something-the-writer-didnt-mean](https://www.latimes.com/socal/burbank-leader/opinion/story/2019-12-05/a-word-please-fused-participles-can-suggest-something-the-writer-didnt-mean) ...although ultimately, I can't say that I agree with Ms. Casagrande's conclusion that which form you use changes the intended or received meaning. I think that's the kind of distinction that grammarians convince themselves of in their laboratories, but which finds little purchase on the steep slopes of real-world usage. My "professional" opinion is that the forms are purely interchangeable, that native speakers overwhelmingly fail to distinguish any difference in meaning between the two, and that increasingly, the possessive construction (once favored by style guides) is becoming little more than an archaic artifact or a pompous affectation.

None of the examples you used is particularly unusual/incorrect

Both are correct, but colloquially, you/me is far more common. I personally only use 'your'/'my' in specific instances for emphasis, that's about it lol

As others have said, both work. However, I think it's useful to say how/why. In "I appreciate your coming here", "coming here" is functioning as a gerund (a noun form that can represent a single or generalised instance of a verb's action). Gerunds can be possessed to mark the subject of the verb. The *-ing* form, however, is also used for the present participle. (We usually characterise the continuous/progressive present as the verb *be* followed by the present participle.) Participles are adjectives. In regular speech, you might say "I envy the person coming here" and "coming here" would be understood as a post-positive adjective qualifying "the person". (This has advantages over using a relative clause as you do not need to worry so much about choosing a tense or having subject-verb agreement.) Now let's consider the role of participles in setting the scene: "(with) my father being a biology teacher, I grew up with a keen interest in science". (This construction mirrors the ablative absolute in Latin.) The whole phrase here ("my father being a biology teacher") acts as a composite noun. You can use such phrases as subjects or objects of a verb: "My father being a biology teacher caused me to be bullied at school; I didn't like my father being a biology teacher."

Both are correct, but using the possessive with this structure is more formal (and *slightly* archaic), and not used most of time even by most native speakers.

“I appreciate you stopping by” is what you’d want to say in 100% of cases. The form with “your” is grammatically correct but way too formal and is more old English/outdated.

"I appreciate you stopping by". "If you don't mind me asking".