Having to slam that Yaw stick to get the bar to move is a little worrying. Almost seems like a pot is bad inside the Tx.
Motor #3 sounds a little funny to me.
In that last video, what the motors were initially doing is normal. It will ramp up two motors if you are going slow and once it sees some yaw, it will ramp the other two to counter the yaw. If you stab the throttle from low to high really fast, they should all 4 spool up the same. I think your computer is a little slow to see this (looking at previous videos you posted) since the bar was jumping all around while you were doing things.
Motor 3 sounds a little weird to me and would be something that I would look into further. Maybe it's not calibrated correctly or maybe it was just a bad motor or ESC. Without having it here, I'm not sure.
That Yaw stick on the Tx though has me very worried. It did nothing in betaflight when you moved it normally. When you stabbed it to one side of the other, then it would move the bar, but only slightly. This is sending up red flags in my mind to never use that controller to fly anything until you know why it is doing that.
The last thing that has me really worried is when you lowered the throttle in that last video. The motor speeds stayed UP and didn't come down until after you kit the disarm switch. This is again a cause for concern.
If you have AIR mode turned on, turn it off and try everything again. You can also try turning off the Accelerometer. This will bypass the gyros (self level) and give you a better feel for what the quad is actually doing.
But keep that Tx in mind, something is not right there.