There are a lot of questions here, I will do my best to answer as many as I possibly can.

PART I

First of all, it is quite important to remember that at its so called 'foundation' in 1265, the English Parliament was very, VERY different from the modern one in size, scope, power, and purpose. That's just to get us on the right page. In no way was English parliament under Henry III (King in 1265) or Edward I (His successor) the primary government organ responsible for passing all legislation and generally moving the Government forward. The King wielded considerable authority in government. But we are getting ahead ourselves. How did the Parliament form?

We must go back quite a ways to reach what I consider a body that holds the origins of the English Parliament. This would be the Anglo-Saxon Witan. Short for Witanagemot. This was a body of 'wise men' whose purpose was advising the King on important matters. The scholarship surrounding the Witan has developed significantly in the last century, so while originally it was seen as a grand parliamentary-style body of nobles/churchmen, this is far more debated. Take this quote from John Maddicott:

But the word carries with it, however unjustifiably, a fustian air of decayed scholarship, and, in addition, its use may seem to prejudge the answer to an important question: do we have here an institution, a capitalized 'Witan', as it were, or merely a lower-case ad hoc gathering of the wise men who were the king's councillors?

In any case, we have numerous records of various English Kings who are somewhat beholden to the advice and opinions of a group of notables who would often be in attendance at his court. This would not be unusual across Europe at the time - it makes sense to consult the various Lords/Bishops across your Kingdom for advice on how to rule it, especially if your decisions can/would affect said people. There is an unwritten rule throughout this period (and after) that a good King consults his Lords/Bishops, listens to their advice and therefore takes a well-worked course of action. I like the way Frank Barlow puts it:

it was impossible, and probably considered improper, for the king to rule without help and advice. Aethelred II, Edward's unfortunate father, was called in derision "Unraed", the King without counsel. But the King took counsel where he chose. there was no constitutional theory

Why then do we have a more formal body (Parliament) that grows out of this Royal circle of counselors? To some extent it is to be expected - it should be noted that the French parlement especially that of Paris, grew in a similar fashion. But a defining difference for the English example comes in the tide-turning document of the Magna Carta, of which I'm sure you've heard.

Between the King's witan or council of the Anglo-Saxons, and the adoption of the Magna Carta in 1216, you have an important period that Barlow terms the 'Angevin Despotism'. It was during this period that Henry II, out of the maelstrom of the Anarchy, established a stable and expansive rule that grew the Monarchy to it's largest medieval extent. His rule encompassed England, Normandy, Brittany, Aquitaine, Anjou and more. In the midst of this growth, Henry worked hard to curtail the power of the Nobility, knocking down castles, expanding the Royal Demesne, and promoting the Royal bureaucracy. While this caused more than some trouble amongst the nobility, Henry II was all in all a strong ruler in terms of suppressing these complaints.

On his death however, much of this falls apart. Contemporary with the rise of Phillippe Auguste in France, the English Monarchy had difficulties keeping together its various holdings, and Richard and especially John were ultimately not up to the task. John in particular had the misfortune to lose much of the continental holdings his family had previously enjoyed, and his campaigns to get them back resulted in economic and military failure.

The Baron's Rebellion that followed in the early 13th century was in many ways a revolt against a weakened monarchial government that had in the past century directly curtailed the power of the nobility and Church. in Barlow's words,

The king's government had been developed to suit the royal purpose alone..it had been extended haphazardly according to the needs of the moment and was subject to no control except the king's; and he was less interested in general welfare than in general efficiency

The key element in the Magna Carta, that we are interested in at least, is the underlying, if crude, attempt to assert that the King would be bound by law in the same fashion that the Barons were - this assertion that there was a law at all that bound the King would in later times the basis of a constitutional monarchy.

King John died before the Magna Carta truly came into effect. His son Henry III, under the Regency of William Marshal (one of the more interesting medieval figures if you get a chance to look him up), used the Magna Carta as a way to gain legitimacy in the wake of his father's death. Through his long regency and afterwards it would have an important part to play. Henry III has in some ways been cast as a weak King, but that's not really in the purview of this question. What is important for us is the note that after he had reached adulthood, Henry III became embroiled in turmoil that would ultimately result in a second Baron's Rebellion.