First you should try resetting the SMC to check the battery issue. Heres the needed steps SMT reset That should tell you if the logic board or the battery has a problem if the battery still won't charge. For now lets assume the battery is known to be good (you bought if from Apple directly correct?)
Next lets run the Apple Hardware Test, here's the instructions on how AHT See if it gives you any errors. If it does report back what it gave you.
To start with I suspect your graphics memory (VRAM) has a problem so you will need to replace the logic board to fix this. If your battery problem is also a logic board issue it should work correctly afterwards. Once you report back the error code/s we'll know for sure.
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Review these steps in the iFixIt section you'll need to follow Logic board replacement In the process you can try checking the display ribbon cable for any damage (cuts or nicks) in case when they replaced your display they damaged it (not likely). I have this exact model and I open and close the lid a lot and have not worn out the cable (unlike other models).
If you open up your system make sure you have an ESD mat and wrist strap so you don't damage anything.
In this model the heat sink is mounted in the lower case so when you pull up the logic board you'll be pulling CPU, GPU & North bridge away from it. Make sure to fully clean off the old grease on both the heat sink and chips. Make sure to get a good quality grease to replace it with and don't over do it.
A replacement board will go for about $400 to 500 (USD) depending on the CPU speed.
This is not the best first repair job one should try but if your game, good luck!