Try updating the firmware. See if that works. If that doesn't work then try cleaning the battery contacts with some alcohol and a toothbrush (carefully). If that also doesn't work then it might be that you got a bad battery. If that's the case, then contact iFixit for a replacement battery.
First, go into System Preferences, then Displays, then click Color. You should have a lot of options listed. Click on the top one that says Color LCD. You should see the colors change. Hope this helps.
First, try resetting the network settings by going to Settings, General, Reset, then Reset Network Settings (this will erase the wifi passwords too). Enter your passcode and wait for it to finish. If that doesn't work your antenna may be broken or got disconnected somehow. If that's the case, you can replace it by following this guide. Hope this helps.
Every battery is going to break down over time. Some batteries degrade faster than others. Maybe you just got unlucky and got a battery that degrades faster than others. Just try to live with it or maybe asking for a replacement(?). Hope this helps.
After you shorted the PSU, the fuse probably popped. Try locating the fuse and replacing it with a new fuse. If that doesn't work, then the other part of the PSU is fried and you have to replace the PSU. Hope this helps.