"Crown", "Her Majesty's Government", "Realm", "Whitehall", and other terms like that all mean the same thing, the entity that runs the UK. The Queen is the Head of State, she formally sanctions everything that happens, but in practice it is the civil servants, the politicians, the armed forces and so on who run the country on a day to day basis.

If the politicians want to pass a law, first they debate it amongst themselves in the House of Commons, where all the members are elected by the common man on the street. The bill is then passed onto the House of Lords, who are an unelected body of politicans, but generally retired former Members of the House of commons as well as former generals, admirals, bishops, dukes, etc plus any other people given a life peerage. The Lords will cast their experienced eye over the bill, correct any spelling mistakes etc then pass it back to the House of Commons who will debate it again, vote on it and if it passed, it will go to Royal Ascent. this means the Queen has to sign the bill saying it is now law. This is how it works here, everyone else does the all the drafting and debating then the Queen gives her seal of approval, she won't get involved in the day to day nitty gritty.

Everything is done in her name, all government property is Crown property, everything from Job centres to Naval dockyards is crown property. Whenever you see "HM", "HM Government", "Royal" and other words like that, it means the UK Government. It's all one and the same.

Yes, there are factions within the Government, you have Labour and Conservatives at one level; at another you have the Army against the Royal Navy or the Army agianst the Royal Air Force or the Navy against the RAF; then you have MI5 against MI6, or the Foreign Office against the Ministry of Defence; or the Police against MI5 and MI6; Oxford against Cambridge; Eton against Harrow, socialists against fascists and so on.

where does the Gulf of Mexico fit in with all this, i've no idea, it's not necessarily which faction does BP belong to, more which faction do the people running BP belong to. I would guess their faction is the City of London, if that is correct, then what is going on in the GoM is probably just good old fashioned greed, just like the Falklands. And the crown will turn a blind eye where possible, because if BP makes a lot of money out what ever it is down there, that means more money pouring into Crown coffers. Lovely Jubbly.