In other words, how long can you be inactive in ranked before you're kicked out of a promotion series?
No, the series only ends once you have played those 3 games. They can be be played over any length of time.
Yes. I don't know why the top answer says you cannot be kicked out because that is false information. Decay is a thing in league of legends which causes you to lose lp over not being active for a period of time. If you do not play for 28 days from getting into your series, you will be kicked out of it. http://forums.na.leagueoflegends.com/board/showthread.php?t=3742674
As of June 2013, promotion series now have time constraints. The series ends after 28 days of inactivity.