I suspect the battery is kicking the bucket. After all, I can say for certain that a New 3DS XL will not show any battery indicators if there is no battery in the device. I would assume that Nintendo would use similar technology in their home consoles. If nothing else, try the charging stand. I use it and it works for me.
You can’t change a home button on a Touch ID device without loosing Touch ID. Be lucky you’ll still be able to use the home button, unlike the iPhones 7, 8 and SE 2.
Restart the device, I had issues with the exact same model of iPod syncing with my MacBook Pro and I found that restarting it a few times fixed the issue. It’s likely due to some timing issues on Apple’s part.
Try pairing/connecting a controller to it. If it connects, it might be a video port issue. If not, it might be a logic board issue. I can’t help with either as I don’t own a PS3 of any sort, but this is speaking as a Wii U owner (where the HDMI port was notoriously prone to failing).
To my knowledge, most MacBooks won’t turn on without a battery, if it’s not removable without unscrewing the bottom case. I speak from experience with a Mid-2014 MacBook Air
(1) Something almost identical happened to my Mid-2012 Unibody MacBook Pro, and my iPhone 6s has a “moody” battery where sometimes the battery is at 80% capacity, sometimes it’s at 20%. The capacity might go back up. (2) I’d say that something is off there… If it works it works. (3) I get ~4 hours out of a new battery for the aforementioned MacBook Pro but I couldn’t tell you. (4) Not 100% sure but errors can occur.
Sometimes iPhones/iPads just do that. iDevices (outside of iPods) will not allow the charge meter to go up so it’s likely the actual battery charge is/was above what it actually was (I have an iPhone 6s that has the same thing, every now and then the reported battery level tanks and the charge gauge goes to like 16% or something but the battery was actually at like 70%)