What happens if BJ simply refuses to comply with the legislation that is being voted today and does not request an extension of the Brexit deadline to the EU?
Can the EU grant that extension even if the UK government doesn't ask for it?
Good questions - and unfortunately it is not all that clear what the answers are.
Technically, the EU cannot grant an extension unless all member states (which includes the UK) agree. Hence the British PM can effectively veto it.
However, if Parliament has legislated on requesting an extension to Article 50, then the Prime Minister would be acting illegally by doing so. What happens then is anyone's guess.
I think it is unlikely to come to that though - Johnson has called for an election, but this is probably going to be delayed by opposition parties who will (temporarily) block it in order to ensure that the UK cannot leave the EU with No Deal on October 31st.
That means that when Parliament returns in mid-October, there will be time to hold a vote of No Confidence in the Government, which will almost certainly pass now... esp. as 21 MPs have just been kicked out the Conservative Party. That gives the opposition 14 days to form a Government, and hence if they can agree on who to lead (at this stage it is likely to be Jeremy Corbyn), then Corbyn, as Prime Minister, would be able to request an extension, which would almost certainly be granted.
Pretty much what
@Tired Tyres said, 'Remain' is split between two parties that can't realistically rule, where as Brexiter's now have a single rallying call.
Under the circumstances, I could see a 'government of national unity' deal being struck between Labour, Lib Dems and the SNP to not oppose each other in seats where the 'Brexit alliance' could win. It is, perhaps, unlikely... but unusual circumstances require unusual solutions.
Note that the Brexit Party nor UKIP have ever won a seat in a General Election. The Tories have also just imploded. They are both pursuing an aim that is supported by around 1/3rd of the UK population, which may be enough to deliver a majority for a coalition, but they stand no chance unless they formally agree to work together... and that is, hopefully, quite unlikely.