Awesome! Congrats, I know what a huge relief it is when it's all finished and it WORKS! Yeah, I just realized that I never told you to do a hard reset. By unplugging the logic board you effectively did a hard reset. Your old LCD might actually be fine! I wouldn't mess with it though if I was you :) I would try and sell the old LCD on eBay or a repair shop around town. You might get something for it.
You shouldn't need to use glue on your iPad repair. There should be adhesive that comes with the repair kit that you buy. Here's a picture of the adhesive map for the iPad 4 Here's another picture without the map. If the adhesive doesn't hold the screen down all the way – sometimes this happens if the adhesive is poor quality – you will want to use super gel. You can find it on Amazon here
You may be out of luck. You can try to manually download the WIFI only version of the software, but I don't think that will even work. I think the best recommendation I can give is that you get a new logic board/modem and swap it with your broken one. Or you can try to get someone to solder the modem back in place. You may be able to find a company online who does this type of solder repair, but the price may be close to the cost of buying a whole new logic board.
That's a hard one. Without being there to see it in action it's hard to tell for sure. Does it make the clicking sound like a smart cover does? If the home button is in a constant state of being pressed down I would unplug the digitizer assembly and then plug it in again. (Make sure the iPad is turned off before doing this) Then test again. If it continues like this then you may have a defective part. I've had high levels of defective parts on the iPad mini. Did you get the part from iFixit or somewhere else? There are also 4 pads in between the LCD and the digitizer that can sometimes interfere with the functionality of the digitizer. These pads can be removed easily. Typically iPad mini's with the phantom touch issue are as a result of these tabs, but this may also be affecting you. Good luck! Let me know the result.
I doubt that the digitizer is what is causing the issue. It's likely something else. The first thing I would do to troubleshoot: Restore the iPad in iTunes (hopefully you have a backup) you may need to place the iPad in DFU mode to get the restore to work. Let me know how this goes!
I think there is a separate volume toggle in settings for the music app. Is it possible that this setting is just all the way down? Seems odd that one would work, but not the other. As far as hardware to replace, you will need a new logic board. The speaker assembly should be fine if you can hear other sounds coming out.
That's tricky. You may have damaged more than just the LCD. Unfortunately the best way to diagnose is to insert a working LCD and see if that solves your issue. You've done a good job trying a variety of things to fix the issue, but It's definitely hardware related. If you post a clear close-up photo of the ZIF connector where the cable attaches to the logic board, then I may be able to tell you if I see any visible signs of damage.
One of two things may be the issue here. (1) You have a shot/dead/defective battery that needs replaced. (2) The iPad power management system is failing. If this is the case you will need to replace the logic board or do some intense soldering to fix the issue. If I was doing the diagnostics myself, I would replace the battery first. Then test again to see if the issue was solved. If you are still under Apple warranty I would recommend having them take care of this issue for you. Battery replacement on an iPad is a real pain. I do it, but I don't recommend it.
In my experience one of two things could be the issue. (1) You have a defective part. (2) There are 4 foam pads that sit between the LCD and the back side of the digitizer. For some reason these pads will at times interfere with the touchscreen causing the digitizer to exhibit "phantom touch syndrome" as I like to call it :) These pads aren't needed and I'm not entirely sure why they cause issues on some digitizers and not on others. Let us know the outcome!