Sounds like you have mirroring turned on. You would do that if you want the external display to have the same image as the internal display (mirrored)
That ALSO means that you would have the same resolution appearing on both displays --- 1280 x 800, which is the native resolution for the MBPro display.
You have to turn mirroring OFF to use a different resolution on the LG ultrawide:
Go to the Displays pref pane, then the Arrangement tab. Uncheck the box "mirror displays" - or whatever that box says.
That will give you an extended desktop on both displays. The Displays pane will allow you to set the display settings for each display independently, so change the display settings to what you like.
Best cable? Get a one piece mini displayport to HDMI cable - such as
this. Get the length that you like.
Can't help much with connection to a PS4. That must be in the PS4 settings somewhere, or the PS4 doesn't support that 2560 X 1080 resolution, or it is another symptom that your present HDMI cable that doesn't fully support HDMI 2.0 or higher, which is what the LG ultrawide would need
I do see that a standard PS4 doesn't support more than 1080P, so your ultrawide display would looked stretched. Nothing that you can do about that, except upgrade to a PS4 Pro. Again, make sure your HDMI cable supports HDMI 2.0 or higher.