I’m amazed this hasnt been answered correctly yet. No! They are not completely compatible. No! You should not replace a 5C battery with a 5S battery. The connector is the same and the 5S has slightly higher capacity, which means slightly longer battery life. HOWEVER the 5S battery is slightly wider which means it either will not fit or will be a dangerously tight fit. They also expand with wear which could lead to serious problems even if it initially fits. Remember: If these batteries puncture they will catch fire and potentially explode. I’ve measured 5S batteries to be between 33 and 34mm wide, whereas OEM 5C battery cells should only be 32mm wide. I highly discourage using 5S batteries in 5C’s even if it initially fits! You do not want to burn your house down (or somebody else’s)!
It could be a number of things: (1) Faulty cable (2) Dead battery (3) Bad battery connection to the logic board (4) Bad dock connector (5) Logic board damage Since your laptop says the device in unrecognized start by trying another cable and then seek professional help if it doesn't work.
You need to get the wifi-module on the logic board replaced or reballed(probably cheaper with a replacement chip than reballing it) Any attempts to heat or freeze it will likely make the problem worse as the chip will only retract further. Chip replacement or reballing will likely be your only solution.
As I see it three things could have happened: 1) You've damaged the power button flex cable. (Most likely) 2) The flex cable isn't properly connected. 3) You've damaged the logic board. (Let's hope not) As for the digitizer, the digitizer flex cable has probably been damaged during install or you've received it faulty.
Doesn't iTunes show you what's taking up your space? (Like music, video, photos, apps, other) You should probably make sure you don't automatically sync it with as many photos/videos as it can hold. You can always try backing it up and doing a restore.
Have you had the power disconnected or rebooted the device since the white screen started showing? Sometimes it just needs a restart to recalibrate the LCD. If rebooting the phone(by holding home+power for 10sec) doesn't work, then the LCD is probably damaged.
The phone also states it is searching for signal although i am able to call out, receive calls and texts. That's just a software issue - It'll go away after a while or after you reboot/turn airplane mode on/off. I think this issue started appearing after iOS 5. On turning the phone back on the power button would not work and still does not work, is it a dodgy contact somewhere? You might inadvertently have broken the power button flex cable although you might be lucky it just isn't properly connected.
Have you listed the correct device?(iPhone 4S) Since the headset jack, vibrator and speaker doesn't work It sounds like an iPhone 3G/3GS where flex cable #5 isn't connected properly or broken. If so - check that flex cable #5 is properly connected and replace the cable if reconnecting it doesn't work. If it indeed is an iPhone 4S I don't see how the issues can be connected - except for extensive logic board damage.