I’ve had multiple people come into the repair shop that I work for with iPhone 6’s, and after updating to iOS 12 they experience the issue you site above. No hardware replacements will fix the issue (I’ve tried it all), as it is purely software based. Get in touch with Apple support and they should be able to give you further instructions towards a fix.
The R4s are a bit touch and go with DSi units, I had one work for two systems, but not work for three others, all causing the same issue. You can try another card, or if it’s easier, another device. I believe it’s got to do with the last few system updates that were pushed out for them before they were left for the 3DS.
For the repair shop I work in, after trying a replacement battery and charge port, if the device discharges quickly we do just as you did, in trying multiple batteries. Assuming that all connections read strong via the lightening port, if all batteries are discharging quickly, then you’re correct in assuming that the chip is likely to blame. More often than not repairing the TriStar is a temporary fix, and doesn’t hold up well over time, hence the lack of an offered warranty. You could get the chip fixed and cross your fingers, or look into a new phone. Best of luck!
More than likely you’ve received a faulty part. Try to get in tough with the provider of the part you purchased about an RMA or swap out, or just order another part. If the second screen is doing the same thing, you may have damaged a connector in the phone during you initial fix.
More than likely, yes. You may need to update some drivers to match the new hardware, but theoretically the update should be possible. If you give it a shot, be sure to post about it so the internet can know too!
In the repair shop I work for we normally spearhead problems like this by popping the phone open and testing out a mixture of new battery and charge port. Sadly, there is not a sure fire way to know if the issue is one or the other until you try both. More often than not the issue is with the battery, though you can’t be sure without testing.