The grey bar and non responsive touch issue is something that happens often with the iPhone 6 plus , i would normaly recommend you go to the apple store but as you mentioned there isn't one near by. Your best Option would be to atleast call Apple Support and see it they will be able to replace your device. If they are not able to replace it then you can always try paying for a Out-of-warranty replacement from apple (Cost is $329)
Heres a link to that support page.
https://support.apple.com/iphone/repair/other