That sounds like it might be a loose connection or damage on the display cable. I would attempt to reseat the display cable and try again. When you reseat the display cable, don't put the iPad all the way back together before testing to avoid needlessly replacing the adhesive multiple times.
Hi Harry, The phone restarting is the one symptom that is less likely to be caused by screen damage, though it absolutely can be. I would attempt to replace the display and see if that corrects the issue. If you're buying parts from iFixit, you can return the part if it doesn't fix the issue for a restocking fee.
Anyone trying to answer this will likely need some additional details. Can you include the model of the iPod? From your description, it sounds like an iPod Touch. I am going to assume this to be correct. You've likely damaged the digitizer, which is what responds to your finger when you touch the iPod. If that is the case, you'll need to replace the digitizer. The fact that you have a white line on the display means you've likely damaged that as well, and will need to replace that at the same time. Once you know the model, you'll most likely find a walkthrough for replacing the affected parts on this site, and probably the parts for the repair, if you need them.
I would inspect the lightning connector. My guess is that there is either debris in the port, or some of the pins are bent/missing. If the former, you can generally clean it out with a small instrument, just be careful not to damage the pins in the process! If the latter, you're more than likely looking at replacing the lighting connector Here's the Guide for that replacement, if you feel up to the task.
This issue sounds like a match for the Paid REP from Apple. You'll pay the cost of a display replacement, but they'll replace the entire phone, as this is an issue with a component on the logic board. I don't know how much time and money you'd end up spending if you tried to repair this yourself, but in the US, the cost of replacing an iPhone 6 plus screen is about $150 through Apple.