Isn't this is supposed to be "you"?

“On your” is correct. It is an imperative demand statement. “I insist on your X” is a kind of fancy way of demanding the person you’re talking to do something. “I need your attention” follows the same format.

Here, “telling” is a gerund—it functions as a noun. Here’s an example of this syntax with a word that’s obviously a noun: “I insist on your discretion.”

"...on your <verb>ing" is an uncommon construction, but I see it once in awhile. I think "...on you telling me the truth" would be grammatically correct, I think, but a bit awkward. "I insist that you tell me the truth" would be more natural.

no, "telling me the truth" is basically a noun in this situation, so it's "your" which of course possesses a noun. But you can say "you" here too, I don't know if it's technically incorrect, but many people would say this with "you", it's very normal and may well be a standard option at this point if it didn't use to be. "your" is probably the slightly more "formal", "less casual" version

It's fine; it's the same as "I insist on your honesty."

The pronoun "you" could work, and it’s what most people would say in this context nowadays. But the more accurate pronoun to use would be "your," which is in the possessive case. The reason for this is that the thing that is being insisted on is the person's act of telling the truth, not the person themself. But once again, this construction with the possessive case is hardly used nowadays as it can sound stilted and formal. To avoid it, people typically use the objective case ("you") or just rephrase the sentence, like so: "As your lawyer, I insist that you tell me the truth."

This is --to me-- correct, but somewhat stilted English. The construction you're looking for, and that I would write, is "I insist that you tell me the truth." You can't really 'insist on' a person in this sense, you would instead insist on someone *doing* a thing, in this case "telling the truth". The meaning is slightly different though, as there's some ambiguity my way whether or not the person tell the truth in just this one instance rather than from that point forward. The way it's written above strongly implies the latter. An important distinction for a lawyer, but not very natural sounding to a layperson.

It's a fancy way of saying something like "I insist on your cooperation" vs. "I insist on you cooperating" vs. "I insist that you cooperate." Likewise, "I insist on your paying attention" and "I insist you paying attention" and "I insist you pay attention." The idea is that there's an intangible thing that is "telling the truth" and that it belongs to you. And that person is insisting you give that "telling the truth" to him.

Either one is fine. I insist on you telling me the truth is more informal. I insist on your telling me the truth is more formal. It’s possibly also more British but then as an American that may just be that I associate these more elegant structures with Standard British English.

you and your are both correct

If you use "insisit + <subject>" this will be in the subjunctive mood. So it will be : I insisit (that) you tell me the truth.

Hey is that destination

Preposition *on* needs an object. That’s the gerund *telling,* (or in this case the noun phrase *telling the truth,* which itself contains an object). If you want to modify that to show posession, you get *your telling.* The way to use *you* would be “as your lawyer, I advise you to tell the truth.”

Using "your" is the only correct version of the sentence. The reason is that "the truth" belongs to the person. Telling (with the "ing" suffix) is a gerund, or a verb used as a noun. So, the sentence places importance on telling the truth. When you place a pronoun before a gerund, the action belongs to the person who needs to tell the truth. That means the pronoun requires a possessive form. Thus, "your" is the only acceptable word to use.

Gerund spotting! "You" reads better to me but this is also correct and, perhaps, a little more fun 😏

I want to see the guidance for the questions because I'm utterly stuck on how Q's 5 & 7 work.

As others have said, "telling me the truth" is a noun phrase, and "your" makes it possessive as belonging to you. Substitute the noun "honesty" and reconsider: "I insist on your honesty." "You honesty" would be obviously wrong.

I insist on your telling me the truth is correct, but only used for very formal occasions (or if you’re a fancy British person). 

Both work. Either You: I insist on [the act of you telling me the truth] or Your: I insist on [your telling of the truth to me] The bottom option would be considered formal most likely.

As many people have pointed out, “your” is technically correct, but this is very formal language and few native speakers would bat an eyelash if someone said “you” instead.

So, this used to be the only grammatically correct way, for the most part. Participles in English were more like nouns or adjectives. So saying "you telling" was ungrammatical. Just like you wouldn't day "you man" or "you big", you also wouldn't say "you walking". (This is why simple participle sentences always use the "to be" verb, as in "you are walking", similar to "you are big"). But languages shift over time. Participles have become more "verb-like", and so they start taking subject agreement as a verb would (in this case, the nominative "you" instead of the genitive "your"). So while you're right that "on you telling me the truth" would be more common nowadays, the older your+participle tends to be seen in more formal settings or in older written materials.

Can someone suggest a website or online platform where I can do such exercises?

A less formal way to phrase it could be "I insist that you tell me the truth"

It’s almost sort of a saying “on your telling me the truth” if it wasn’t a saying a native speaker would probably say something like “I insist that you tell me the truth” “I insist on your telling me the truth” is grammatically correct as long as we use “telling” with “your”. “I insist on” on what? “Your telling” your telling of what? “The truth” or specifically “me the truth”

As many have pointed out, “…you telling me the truth” would fine, especially in a less formal context. I would like to add that it can be fine in a formal context, as well. It suggests a slightly different intention on the side of the speaker though. I would expect to hear it in a context where the speaker is contrasting ‘you’ with another person, i.e. I want to hear the truth from _you_ and not from someone else. The version “…your telling me the truth” is more neutral, i.e. “don’t lie to me”.

Insist on + noun or noun phrase. "Telling" in this case is a gerund, which is a noun, which is why it takes the possessive adjective "your", not the pronoun "you".

your