Make sure your motors and ESCs support 4S, that will likely make it take off better, but you will need to tune your PIDs, P will have the biggest effect.
I don't think that is possible with most LiPo chargers. Check the voltage, it may not have charged at all or went bad due to something like over-discharge
The direction of the motors is not determined by the data cable but the three phase coming out of the ESC, swap any two of the three wires and you will reverse the direction of the motor.
The three phase side is the part marked "Bullet connectors"
Just watched your video, that is certainly not a COG problem. Are you using the correct props along with the motors spinning the right way? It looks like you are either using CW props on CCW motors on the front or they are spinning the wrong way.
Place two fingers under the quadcopter along an axis you want to test. If it balances good, if not you have to move something like your battery around.
When turning the quad on is it facing away from you? Though if you plan to get into bigger quads I suggest you avoid headless mode as it teaches bad habits.
In that case the first motors would obviously provide you with more power, and better efficiency if I'm not mistaken. You should be able to get some impressive flight times with them.
Seen a lot of gopros mounted to 250s, but you are going to want that protection plan hahahahaha. When you go to the store to get a replacement be sure to tell them how you keep falling of your bike ;)