It sounds like your home button is damaged. I suspect the reason that it appears to function on the lock screen is because the Touch ID sensor is actually doing the work, not the home button itself, then once the phone is open there is nothing more for the Touch ID sensor to do and the failed home button does nothing. I’ve seen this happen before when the home button is not treated delicately during a screen replacement. Because the home button is specific to the board the only way to restore functionality would be to bring it to Apple to have them replace it, or purchase and install an aftermarket home button that can connect to your phone via bluetooth (if I recall correctly). Hope this helps!

If your iPhone battery is low enough it may not show the red battery symbol while it is charging. Best would be to let it charge with the OEM cord enough to power on and then test your other aftermarket cables. A note on aftermarket cables: cheaper cables often lack the hardware to prevent voltage spikes which can damage your device, so it is best to stick to OEM or high quality aftermarket cables. Hope this helps!

It can be replaced, but it isn’t easy. The camera housing is spot welded into the metal part of the housing so you have to cut it out of the frame. Attaching the new housing also has to be done with some kind of super glue so it binds tightly to the metal. Hope this helps!

Take a look at the wifi antenna near the top of the logic board. Any issues with this antenna will greatly reduce iPhone’s ability to detect and connect to wifi networks. Hope this helps!

Best way would be to just use the phone until the battery dies

It sounds like your phone is in DFU mode. To take it out of DFU mode simply hold the power and home button down together until you see the Apple logo, then release both. If your phone reboots successfully then you should be able to make a backup of your phone and save your data Hope this helps!

The home button needs to stay with the original device, other than that you can interchange parts as you choose

It sounds like the board may have been damaged due to the drop. I would try opening the phone and looking for anything that doesn’t look right, cables amiss, stuff like that. If nothing looks wrong it could be board damage. Hope this helps!

Aftermarket batteries will often do this, even occasionally causing the screen to flicker around the edges (especially when the phone is low on battery). If it doesn’t get any worse I would say it’s just due to the battery and shouldn’t really be any cause for concern. Hope this helps!

It sounds like it could be the LCD starting to fail. I would try testing the phone with a known good screen and seeing if that solves it. Hope this helps!