I really hadn't intended dragging this league stuff out into three parts, but that's precisely what's happened! There are a few other facts – and little quirks, simply associated with league matches – that you may find interesting.
I know that some leagues in the U.S. do this, but it seems that very few rely on an open draw for the evening's games; many rely on the captains choosing who plays who, usually dependent on ability. In Britain, most draw for opponents, although sometimes they play “pairs (or fours) across the board”. All that means is that the pairings are based on position within the team - 1 plays 1, 2 plays 2 etc. This is most common in fours and pairs, and is rarely used for singles. It's called “across the board” as the teams are invariably written up on the chalkboard.