Is the silver line on the outside of the screen like a fingerprint would be? If so, try wiping it with a soft, clean cotton cloth (like a lens cleaning cloth for eyeglasses). If not, then it is possible your LCD has been damaged and would need to be replaced.

It sounds like the LCD is broken. Laptop LCD's are relatively inexpensive to buy and relatively easy to put in. If you have the right tools, it would be no problem to put one in yourself. I would definitely watch a repair video first if there isn't a guide on ifixit.

This isn't super helpful, but one that I was working on had this same issue. I left it on in its boot loop while taking it apart to diagnose the issue, and it snapped out of the loop as soon as I took out the topmost screw of the plate holding down the battery connector. Again, it's not super helpful but it leads me to believe there may have been some sort of debris near the battery connector causing a short or something. Be sure to clean off the motherboard in that area and the area near the flexes for the display connectors.

There may be some debris stuck in the headphone jack. Try air dusting it out first, and if that doesn't work you may need to replace the headphone jack. The guide is in the link below. Good luck! iPhone 4S Headphone Jack Guide

Did you replace the physical button or the flex cable that sits underneath the home button? If you haven't replaced the home button flex cable, try that first. If you have, check to see if the metal shield that the home button is attached to isn't warped to where the contacts on the button of the flex aren't contacting the motherboard.

This usually works a little better than rice. If you want something more effective, disassemble the phone and clean all the contacts and the mother board with a toothbrush and some isopropyl alcohol. Get all of the buttons and the charger port too, just to be safe. There's no guarantee, but if there's any corrosion that should help. Good luck!

Sometimes that will occur if your date and time are not set correctly. Occasionally when the battery is disconnected, the phone will stop automatically updating the date/time correctly. If they are incorrect, correct them and restart your phone. Good luck!

I had the same frustrating problem a few weeks ago. After a few hours of fiddling with the back cover and trying some software resets to no avail, I rechecked that all of the various flex connectors were securely in place and swapped the back out with one on another phone and it worked fine. I'm not sure what fixed it, but I have a hunch that the the standoff screw that the battery connector screws into might have been too lose so that the battery connector didn't fit all the way into place (leaving the connector high enough off of its socket to be affected by the back cover).