1) Does a quest need to be completed in a single Event, or just a single Block? So I would need to play 3 matches with an arena in one evening to get it?
2) If it is a single event, can I complete the quest again the next week, if still in the same block?
3) Block 2 has some similar events as Block 1. If the answer to number 2 is no, can I complete these quests again in Block 2 if I completed them in Block 1?
1) Single Event. You can't keep a "running tally" of your progress toward the Bonus Objectives from one Event to the next.
2) Yup! You can potentially complete each of the current block's Quests once at every Event you attend.
3) Not relevant, since the answer to 2 was yes.