1. I am not sure when you are allowed to play cards and use abilities.
That's pretty hard to answer. Can you say which abilities you are referring to? Some can only play during a fight, some can only play NOT during a fight, some are played on your turn, etc. If the ability or card text doesn't say, then you can usually assume it can be played at any time.
One general rule for all Flying Frog games is that you don't have to guess whether you will need a card -- they let you roll dice first, and afterwards if you decide you need more dice, you can then play a card. There is one exception: a card that cancels another card has to be played before the second card causes dice to be rolled.
2. Does the Shadow Track move one step closer to darkness if an Evil Elder is killed?
The Shadow Track does not move when an Evil Elder is killed. See the designer's post here: An Evil Elder is no longer considered a Town Elder.
By the way, the rule for Town Elders is that the shadow track goes down by TWO when they die, not one.
3. Can you play event cards at any time during the game or only during your own turn?
If the Event says to play it "as an action", then that means you can only play it on your turn during your actions (which is after you move/linger). If it doesn't say "action" on the Event, then you can play it on anybody's turn or during the Mystery phase.