If the UK government doesn't ask for article 50 extension, can parliament do it instead?

Either side can apply to extend Article 50 of the Lisbon Treaty (2009). This means that in exceptional circumstances, where there is clear potential detriment to both the UK and the EU, the EU could if it was politic and appropriate so to do, to enable a pending general election to first take place following a vote of no confidence in the Government, apply to the UK Government to further extend Art. 50.

A cynical postponement of the general election by a failed government diametrically opposed to the will of Parliament, while the UK is still at that time a member of the EU could contravene the British electorate's freedom of expression, contrary to Article 10 of the European Convention on Human Rights. The critical muting of a nation's voice goes to the heart of the Convention's raison d'etre to preserve notions of freedom and democracy. The dubious legitimacy of any such precipitous political skull-duggery would be challengeable in High Court via the Human Rights Act 1998 and on appeal in the Supreme Court.

Where Parliament passes a law requiring the PM to seek an extension, the refusal so to do by a lame duck British Government would be a contempt of Parliament and blatently contrary to the Rule of Law.Apart from a vote of no confidence,this would inevitably involve the courts and a writ of mandamus would be issued requiring the action be carried out in default there would be a power to imprison Boris Johnson. Further, failure to comply would give locus standi to any citizen to apply to the High Court pursuant to the provisions of Article 41 of the European Charter of Fundamental Rights to protect the fundamental right to good administration. This includes the right of every person to be heard, before any individual measure which would affect him or her adversely is taken.

If Boris Johnson and his cronies played fast and loose with these underpinning rights, going forward where would that place the UK in the general scheme of international diplomacy? Our national integrity and kudos would be shot to pieces. Who would we be to claim the moral high ground on questions of human rights?

Parliament is sovereign.

They could pass legislation (through both houses) removing the ability of the government to conduct treaty negotiations with the EU and vesting this in a person or commission appointed by that legislation.

(Legally, the UK would have changed its form of government. Parliaments have done this before - the Fixed Term Parliaments Act is an example as is the legislation passed by various Commonwealth parliaments (with similar constitutional arrangements to the UK) becoming a republic, just as a for instance).

The EU can't really help here. The law in the UK is that EU membership ceases on the 31st of October, which means the treaties are no longer in effect and for example the EU courts have no jurisdiction, Freedom of Movement ends etc.

The EU can't really do anything to prevent that. They could possibly make offers, such as allowing a new government (in the event of a General Election) to immediately re-join under current terms, but actually extending the current membership does not seem to be within their power to do.

The correct thing to do in such a case is to call a vote of no confidence, and topple the government. Parliament can then give it's confidence to someone who will make that request.

However, given that parliament taken as a whole has shown an aversion so far to actually voting for anything definitive, rather choosing to just put spokes in the wheels of other plans, it is debatable if they would go down the correct route or just panic at an even later stage.

Probably not.

Article 10 of the Treaty of the European Union states that:

Member States are represented in the European Council by their Heads of State or Government

It follows, therefore, that any notification made to the European Council must come from either the relevant country's Head of State or Head of Government. In the case of the United Kingdom, it would have to come from either Boris Johnson (as head of government) or HM The Queen (as head of state).