Several problems with my iPhone 5S

Today I bought an iPhone 5s from someone who wasn't honest with me. Turns out the LCI indicators are pink, which means water damage.

She told me that the home button doens't work and that the phone shuts off at 40%.

The phone turns on and works 'perfectly' with a few flaws.

Problems:

- Home button clicks but doens't work. Touch ID works and I can get the control center to show up. I can also use the home button to get into DFU mode. Really weird..

- When the phone is turned off and I plug the charger in, it goes to the Connect to iTunes screen.

- When phone is turned on and I plug the charger in, it will charge. But if I afterwards turn off the phone with the charger still in, the phone will turn off and go to the Connect to iTunes screen.

- If I restore the phone in recovery mode, it will restore it and ask me again to connect to iTunes to restore it. Sort of a loop. To stop the loop I plug the cable out once it restored. If I then turn on the phone, I get to the white 'Hello' screen and can normally install the phone.

- When I press the sleep/power button for a long time, the phone turn off, apple logo shows up, phone turns off. (don't know if this is really a problem or me being paranoid lol)

I hope someone very experienced can help me :) I love all input ! Long live iFixit !

Can't get weirder than this.. So the original home button was clicking and Touch ID worked fine, and I could do hard resets, DFU restores with it, but it wouldn't do anything when you click on it after booting when you're on the 'home screen'.

Just installed another home button I found from anther 5S. This one seems to work fine, also when you're on the 'home screen'. A bit sad, because this means I got to say bye to Touch ID functionality. All other problems seem to be gone after replacing the home button. No more problem getting stuck in endless recovery mode loops.

I'm wondering, is there a way to save Touch ID functionality and what could cause the problem with the original one?

Could a dip in isopropyl alcohol do magic? I'm not giving up on the old button just yet ! Why on earth would problems like recovery mode loops get solved by replacing a home button.

Big thanks to you guys Giga and Stijn !!