The British "Constitution" is unique in that it is literally unwritten. There is no single formal document one can refer to and say that it contains the UK equivalent of the US Constitution. So in the UK, everything is done "by convention", "because it's always been done this way" -- that's an oversimplification, and obviously over time things have changed, but you get the idea.
"Prime Minister" is not literally a paid position, but instead IIRC the PM gets paid as "First Lord of the Treasury". Ah, Wiki shows IRC
http://en.wikipedia.org/wiki/Prime_M...United_Kingdom
As the title of of Jeffrey Archer's book "First Among Equals" suggests, historically the PM was literally just that, the nominal leader of various Cabinet Ministers who were all "equal", but as time passed the PM has accrued more power to his office.
Does PM have to be an MP? Yes in that No, he could hold a peerage and sit in the House of Lords, but I don't think there's been a Lord as PM for a very long time now. And today it would simply not be "politically correct" for an unelected lord to hold the premiership. (Checking Wiki, the last Lord as PK was Alec Douglas-Home, Earl of Home, who ironically was an MP when younger (his father was still the Earl and therefore he was not disqualified yet), entered the Lords on his father's death, became PM then renounced his earldom to stand for election to the Commons, precisely because a Lord as PM was considered impractical.)
Opposition leader? There is one (today, David Cameron, leader of the Conservatives), and indeed he/she gets paid a formal salary as "Leader of Her Majesty's Loyal Opposition", and leads a "shadow cabinet" of opposition spokespersons who's job is to oppose a particular minister (so for instance there is a "shadow Foreign Secretary", "shadow Home Secretary" etc), but I don't think he/she has any real influence on legislative agenda. By definition (save for the unusual minority governments), a PM's party has a majority in Parliament and doesn't need the opposition's votes to run the country.