I know English armies in this period were composed mostly of archers with some men-at-arms (and perhaps billmen?), and I presume the men-at-arms were all more heavily armoured than the archers, but how well-armoured were they? Would they all have had complete harnesses or was that reserved for just the wealthiest among them?

I'm also wondering if weapons like bills and falchions (by "falchion" I mean something like the Wakefield sword) which I associate (perhaps incorrectly) with common soldiers were ever used by men-at-arms? And likewise if more "knightly" weapons such as longswords were used by archers or other common soldiers.

Correction: men-at-arms, by definition, could afford full armour. Not necessarily full plate armour. The language of many muster regulations and military contracts hint at this -- "armed" was a common term, "fully armed at all points" or "with full harness" were very common too, but "with full harness of plate" seems to have been used mostly in a descriptive manner for individual men-at-arms rather than as a prescriptive requirement for the class. This usually left considerable room for variation as long as the man-at-arms in question was covered from head to toe in some form of armour. Of course, the meanings of terms and military norms shifted over time and by the 16th century there does seem to be an expectation that a man-at-arms should be able to afford a full harness of plate, but I'm not so sure that it would already have been the case by the Wars of the Roses.