I can confirm that the Google Home Mini works with a simple micro USB input, just like a smartphone. The plug is a bit recessed so not all USB cables will be compatible, but I easily found one that is. You should get a high quality cable because the voltage drop at full volume is enough to make the Home Mini reboot. The included AC adapter is rated at 5V and up to 1.8A, so get an USB adapter rated at 1.8 or more amps.
Maybe your cellular antenna is not perfectly connected. You can try to disconnect and reconnect it : it's the small connector on a black cable near the battery connector. Step 12 of the display assembly guide : iPhone 4S スクリーンの交換 Also degrease the pressure contact with windex to avoid interferences
The screen are the same, wathever the memory size. Screen are different only between models, from top to bottom iPhone 4S, iPhone 4 CDMA and iPhone 4 GSM : http://blog.webtito.be/wp-content/uploads/2012/05/iPhone-Display-Assemblies.jpg
I think you broke the very fagrile glass that holds the LCD. When it breaks the electrical traces break and the screen generally looses entire columns of pixels. You will have to replace the front panel : - Part: iPod touch (4th Gen) Screen - Guide : iPod Touch 4th Generation Front Panel Replacement
Usually I don't remove the last six screws that hold the screen. I just loosen up them a bit. You might have to shake a bit when replacing the new screen but this saves the frustration of putting back those screws with these tiny rings.