You have two distinct problems:
You've already diagnosed it - 950 cycles and the service battery indicator pretty much says it all. From 2013 on, you'll have to replace the top case because Apple glues them in, but the battery can be replaced quite inexpensively.
The flickering can be caused by a number of things like a logic board issue, a faulty connector, a bad LCD, etc. I would try running Apple Diagnostics by holding D while booting from a powered off state to see if you can get a diagnostic code however, chances are you're going to have to take/send it in for service.
A battery, generally, will not cause issues like this because it doesn't dump "raw" power directly from the battery to the logic board. It actually goes through some circuitry to "level" everything off and split it out into the correct voltages the system needs (12V, 5V, 3V, etc). If the battery was causing the problem, you'd see it everywhere on your Mac, not in just one place.
That said, to diagnose properly, you would disconnect the battery completely from the logic board and see if the problem exists on power only. If it shows up, you've got your answer.