I too reconize this as a battery problem. The connectors look intact. If you can fin a battery with similar part no. try it out. Note that there are reports of different batteries to different iPhone 4.
Hello The home, back, search is a seperate part that is wedged in between digitizer and lcd upon install. I have done this install myself. In the bottom of the screen, there is a small gap, that allows for a paper thin Flex cable, to slide in between the digitizer glass and the frame that holds the lcd. My best guess is that this bottom part is damaged or has been disconnected during first replacement.
Hello Sounds like you finally broke the lcd and digitizer. Pink and blue lines on lcd is a typical sign of failuer. Also when glass is broken, moisture can find it's way into the phone. I recommend replacing screen and digitizer.