Unplug all cables except the power cable, turn it on and see if you still have the green line. If without any image input, the line is there, you have a faulty LEDTV. If it goes away, it's the cable/connector. Report back with your findings.
The image itself does not present any significant damage if any at all, so you should be fine just replacing the digitizer. From the (veryveryvery quick) research I've done, the LCD and Digitizer are separate and not glued together as in some models, so the digitizer alone can be replaced, it seems. There are quite a few tutorials on YouTube and Google on how to replace it. Let us know of any other questions or issues that might come up.
There are a few things I'd like you to try. First, pull the battery out if you can, and plug it in. Does it boot? Can you boot into recovery? If there's any SD or SIM card, pull those out and try again. Pull out the battery and leave it out for a good amount of time (couple of hours, to try and clear the cache). Fully let it charge without trying to turn it on, if you can. Then unplug it and try again. I've heard of some bootloops after battery dying caused by software, corruption of some files or partitions when the battery died. Those can be fixed. Best of luck, report back with any developments.
Seems like it could be one of many issues, or all of them combined. Does the device show any signs of charging when you plug it in? An LED or maybe it vibrates? Leave the phone in rice, like the above comment suggested, and after that (ONLY AFTER, in case of water getting into the device, plugging it in can short the components) leave it charging for a while. When batteries go to 0% (which they never should) they take a while to calibrate and get back to the correct voltage. Also, when you do plug it in (a couple of hours maybe, with a good power brick and cable) make sure to check regularly for unusual heating or even sounds. Please report back with any developments.
That's really up to you, to be honest. There are many things to consider. Was the laptop still running fine before? Do you need an upgrade? Are you certain that the issues were caused by the water damage? How confident are you in repairing the battery? No one really knows if it's worth it or not, except you.
Can you please try https://support.xbox.com/en-US/xbox-360/console/check-power-supply and https://support.xbox.com/en-US/xbox-one/console/console-wont-power-on Seems like a cooling or stable power to the PSU issue. Report back after you've tried those methods.
Can you try this? Press and hold the Power and Home buttons simultaneously for about 15-20 seconds until the white Apple logo appears; it can seem like a long time but, don't worry, it will eventually. The iPad will then reboot. From http://www.ipadforums.net/threads/grey-screen-on-ipad2.28804/