First thing: Do not plug it in and leave it unattended. Also, do not turn it on until you have inspected it.
You have 2 separate issues: Lines on the screen and overheating. Lines on the screen could be as simple as the display connector got knocked loose. Open up your iPhone using the teardown guide on iFixit. Disconnect the battery before you do anything else. Then check the display connectors to make sure they are seated properly. I'd also clean them out with 99% IPA just to be sure. While you have the battery disconnected, inspect it for bloating, tears or leaks.
If the battery looks good, then its time to inspect the power ICs to see if one got damaged or knocked off in the drop. It take some work to replace them but its doable. Once you get your power issue fixed by either replacing power ICs, the battery or both, put the screen back on and see if you can power up. If you do, then check the screen for lines. If the lines persist, you know you need a new display.
But please...disconnect the battery before you connect it to a charger or turn the phone back on. LiPo fires are not a joke.