Answer to "home button not working after battery replacement?"Yes the flex cable or one of the connectors in that area to the screen is being pinched down to hard by the metal cover above it. Loosen the 4/5 screws depending on your model that covers the cables and you should be fine.
Answer to "home button not working after battery replacement?"The metal cover above the lcd cable is too tight. Loosen it and then you should be okay.