I would like to respond to your 2nd point on what the Holy Spirit’s conviction proves, and by the way, I believe that the 3 things the conviction proves are in that point alone.
In John 16:8-11, Jesus not only gave the three-fold ministry of the Holy Spirit’s conviction, but He also gives further insight into them.
Firstly, He says “of sin because they do not believe in me”. This tells us that the Holy Spirit uses their unbelief to prove the gravity of their sin. This conviction is necessary because, on their own, the unbelievers do not believe in Jesus.
Secondly, He says, “Of righteousness, because I go to my Father, and you see me no more;” to my mind, the fact of Jesus’s absence means that the one who gives this righteousness would not be “visible” enough to convince them. Therefore, the Holy Spirit through the ministry of conviction, uses the triumphant work of Christ to prove the availability of righteousness.
Lastly, Jesus says “Of judgment, because the prince of this world is judged” This simply reveals that the conviction of the Holy Spirit will use the defeat of Satan, who is the Prince of this world, to prove the solemn certainty of judgment.