Hi, a lot going on here so try these out: First, that looks like a case of your Pro Controller's joystick drifting upwards a little. (Perhaps try calibrating it in your Steam Controller settings?) As for the buttons being swapped, You should either change to the "XB" layout in options or just go to your Steam's controller settings and disable "Nintendo Button Layout" there, as when that's on it swaps the ABXY buttons to match games that only have Xbox button prompts. Not sure why your DPad isn't working, but I tried using the same controller as you and got proper results with my Steam Controller Settings setup like this. I hope this helps!
