Macbook Pro can't wake up (resume) after the battery dies
Two unrelates hardware issues - sounds like a hardware repair might be the logical first assumption.

Also - tearing is a performance issue, so without sitting in front of three of the same hardware, it's hard to know if yours is different than the pack. It could be your expectations are higher than the hardware can perform or it could be a really bad glitch and something that's clearly defective. I've seen both in the past, so I'd focus on getting the hardware power logs and the start up chain / logging / issue and the rest might be a moot point if the SMC isn't starting the hardware reliably. Assuming a problem there, whatever repairs that will replace the GPU/CPU/SMC in all likelihood.

Lastly, portable Macs do not have an automatic "power on" after electricity comes back on, so you could test that the Mac is still functional after a low battery event by letting it run down and schedule a start up or wake for a time when you know you will have it plugged in to power after it shuts itself off. If you're close to an Apple store, have them do the service, but if you really want to triage this yourself, that's how I might start figuring what's up with the SMC and power state after a low voltage shut down and reconnect to power.