iphone 6S black screen after drop -- can i retrieve the data on it?

I dropped my iphone 6S and came up a non-responsive screen with various hues of black vertical lines. I'm able to receive texts and calls on my watch but the phone screen does not work at all. I tried hard reset, and connecting to laptops with various versions of ios to back up to no avail. As soon as I connect the cable I get a haptic response and then nothing. it sometimes recognizes the iphone but says "There is a problem with the iphone" and tries to update or restore (which means loss of all data).

My questions: 1) Is this a common problem? The phone dropped about 3 feet, screen is *not* cracked in any way, and with some combination of reboot options sometimes I get a light background (and kind of see silhouettes of some apps) with a wide vertical black steak but nothing else. I can actually send and receive texts and calls using my watch which is paired to the phone. I get notifications on my watch. That tells me the phone is working but there must be some hardware issue that occurred upon contact that disabled the touch screen altogether. I'm wondering if it is fixable in any way. I've called local repair shops, I think that they are able change cracked screens, but I'm curious if hardware issues like those I'm experiencing are resolvable. 2) Is there anyway to back up the data on the phone before taking it in for repair, just in case the data gets damaged during the repair? I've tried going into itunes to "reauthorize" (was suggested on a different thread to a similar question), however that just reauthorizes the computer and does nothing to back up the phone. Ultimately, itunes sees the phone but cannot seem to access its data. That makes me think there is a secondary issue other than the screen connections being loose or something.