A new logic board is probably going to be your best (only) bet on that. If you can fix that kind of damage then you are more skilled than I am.

Other answers here are spot on. Usually some clear packing tape will do the trick though.

Definitely a faulty new connector. If you short the prongs and get nothing, there isn't much else to explain that. I will say that while this may or may not be true with you, I have "fixed" many lightning cable connectors bu simply cleaning out the port itself. Pocket lent and other dust like to accumulate and pack down inside, preventing the cable from clicking in all the way. you can use a paper clip or if you have some pointy tweezers, and just scrape it out (Careful just to scrape the "bottom" of the hole and not the sides). But if that doesn't work definitely see if you can get the new port replaced from wherever you ordered it from.

I have on several occasions been able to bring a board back from liquid damage, but never an LCD or digitizer. If that is not working after liquid, you're going to need to replace the screen. I'd also recommend taking the board, battery, and charge port out of the enclosure and inspecting it for any corrosion or residue. Something is there causing the heat and once any corrosion eats through the leads or traces, the board is pretty much gone unless you or someone you know is very skilled at micro-soldering (Very expensive to repair, usually not worth it for most people and its hard to find someone who really knows what they're doing.) I hope that helps.