The October date by the EU was not choosen on a whim. The new commission takes over the next day. So no the current commission cannot do that. They and normal people would see it as unethical by tieing the hands of their successors . I suspect that should there be a groundswell from the UK ..(extremely unlikely), they might hold informal meeting s with their successors,and on that basis make it known that a request would not be rebuffed.
No I think the 31st date is fixed, unless there is a quick No Confidence Vote.
The EU can't change anything unless we let/ask them.
We are gone on the 31st unless they come back and renegotiate their obviously renegotiable position.