How does one's start time effect the results of the weekend tournament . Say I wait until Saturday..

From what I can tell, you get placed in singles tournaments based on when you complete (or maybe start/continue) the first puzzle during the tournament period (daily puzzle possibly excluded). I have personally started late Saturday evening and always end up in a fresh tournament, with hours of play ending Sunday evening (fewer hours play).

For team tournaments, I believe when the first teammate completes (or maybe starts/continues) their first puzzle, the team is placed in a tournament.