The iPhone 6 Plus LCD FPC looks like a match. Can let you know if it works when I know. EDIT: IT WORKS! The iPhone 6 Plus LCD FPC is confirmed to work in place of the small 36pin iPad 6 Digitizer Connector. Bless Apple and their cheap recycling hearts.
It sounds like a damaged LCD. I would take that in to be serviced or, if you're feeling capable enough, you can replace that screen. I would order one with a midframe as it makes the entire repair a lot easier. Just make sure to take your time. If you're feeling like you wouldn't be able to successfully do that, any Samsung certified repair center can replace that screen with an OEM replacement and have you up and going!
This most likely means one of the speakers has blown and would need to be replaced. They’re fairly easy to disassemble but unfortunately I haven’t but been able to find a replacement speaker for the system. You might want to contact JBL in case they can service it.
What operating system are you running? Anything new than 10 will likely tax your phone's hardware much more than previous OS's would. It running hot while playing games is likely just the hardware having to run harder, especially if the games are newer. The battery drain is tied in with hardware that is running harder. As far as solutions go, I would recommend maybe closing everything else out while you play those games to reduce background usage. It could be time for an upgrade though.
The water probably damaged the digitizer. I would disassemble it (if you're tech inclined) and dry it out that way. Unfortunately rice doesn't do much to actually pull water out, or prevent corrosion from forming. If the device is turning on though, that's a good sign. You might just need a screen replacement (new Glass/LCD/Digitizer). I've had that fix very similar problems.
I would look for a tear in any of the cables. Specifically the Home Button to Logic Board cable that is with the LCD/Digitizer connector, and the front facing camera connector. I also surely hope you've tried the good and faithful hard reset.
Unfortunately it doesn't look like that part is widely available as of this posting (as the device ages, the parts may appear). Anywhere that isn't using Samsung official parts and adhesives will definitely return to you a device which is not 50m water resistant. It will likely rival a replacement cost.
The charging issue is likely related to the dock. It's a super easy component to replace and test with. If that doesn't solve your issue, it could easily be an old battery failing. A little more tedious to repair, but definitely doable.