Contrary to popular opinion the only different between the two sides at Stirling were the size of forces and the colors of their surcoats. We're talking mail armored professionals for the most part. The Scots were outnumbered 6:1 but made good use of terrain to only fight a third of the army (who got absolutely stomped with like 100% casualties). Sir William Wallace was a knight, his father was a knight, Sir Alan Wallace, and I suspect his grandfather was a knight although I'd have to look deeper. Wallace is Scottish for Welsh, because his family originally came from Cumbria, which is to Scotland what Cambria (Wales) is to England. Stirling Bridge in the movie is based on Bannockburn which occurred nine years after Wallace's death and obviously in which he played no part.
Basically what really happened was after a series of premature deaths the Queen of Scotland was a 3 year old Norwegian Princess and two noble houses, the Bruce and Balliol families were basically going to murder the
out of each other to get it. Naturally the regency council was like "they're probably going to kill everyone we should do something." So Edward of England offered to recognize Balliol (who had less of a claim than Bruce or the Princess) to avoid bloodshed. However he demanded that the King of Scots officially become his vassal as had already basically been the case, but never formally. However this got really awkward so they reorganized and basically broke their word and made an alliance with France which pissed off Edward because they back stabbed him. So he invaded and pressured Balliol to resign and go into exile in France and effectively became sole King. This led to several Scottish rebellions (i.e. Knights and Barons, not peasants) such as Wallace's and Moray's. Ultimately Bruce entered the fray, defeated the English and took over the place and became a national hero. Edward who acted reasonably and fairly throughout (if in his national interest) is considered a villain.
Falchions are designed to bite into mail armor (or at least damage it and beat the bejeezus out of the guy wearing it) so yes I imagine they had some. They definitely existed by the 11th century and this was almost the 14th century. They did not have Falchions during the 6th century (Tristan and Isolde are historical figures) you are correct. I'm not aware of any weapons like that in Sub Roman Britain. They should be armed with Spathas.