Answer to "Apple logo then blue screen"Is just the screws. I had the same problem. Just look at it: http://www.iphonebits.co.uk/iphone-5s-blue-screen-solution/ Everything you need is there.
Answer to "Microphone and home button does not work after battery replacement"I have the same problem. Just after battery replacement. Microphone works on vídeos or calls using speakers, but doesn't work on recording notes. Home button doesn't work too, but Touch ID works perfectly.