It it the horizontal sync circuit. I can be fixed but is takes some time for a professional to get to and replace the sync transistor. May not be worth the money. Good news and external montior should still work.
If possible dry the phone out for a couple of more days. Sometimes blow warm air on the wifi cable. This is because the cable acts like a wick and suck the water up the length of the cable. Your phone sounds like it can see the wireless beacon and connect, but can not transmit due to water being in the cable. You might need to replace the cable.
Definitely check the headphone connector, but don't forget to check the speakers. Maybe the are just clogged with dried ice cream. With the phone off, inspect and clad the speakers carefully.be sue is is completely dry before turning the phone back on.
Take it back and have them check the antenna cable. Sometimes they don't get seated properly after the screen is replaced. Did you check the SIM card. I usually clean the gold contacts on the SIM card. Your provider can also check the SIM card.
Always buy a new battery. A lot of ebay batteries are used. I also would check the refresh settings under General. I had alot of news apps as well as Linkedin and they were using a lot of battery because I was constantly getting push updates. I turned it off and now my battery runs a full day.