For network unlock most can be done by code but finding an honest source is tricky. Here in the U.K we contact the network to unlock and they either unlock free of charge or for a fee, if they refuse then we use a source. If you need the source let me know.
Download ODIN Samsung flashing software. Make sure you also download the correct ROM for your phone. Then connect phone to pc with software on and flash it this will fix all software related problems. If it doesn't you will have a hardware fault.
I would try another screen as some repair centres buy fully assembled screens and just swap home button I.D. It might be a defect cheap part on the screen instead of them swapping the genuine parts over to the new screen