Gosh, this rulebook...
It could be organized better in some areas, I'll give you that. But all of your questions are actually answered in the rules.
Question 1: The player with the map goes first. Who goes second? I assume it's clockwise but it isn't covered in the rules.
Page 6, paragraph 2: "Play continues clockwise until each player has taken one turn..."
Question 2: Let's say I make a successful encounter roll on the first try. Can I reroll the remaining dice? Nothing in the rules or the two examples confirm or deny this. Only something vague about removing dice from the board.
A common misconception is that you are rolling against the encounter, even though nothing in the rules says that. You are actually rolling to get dice that you want to spend
. You may choose to spend those dice to overcome an encounter and/or gain tokens as designated on the Scenario card. So, you must roll (and re-roll as you desire) until you are satisfied (or out of rolls). Only then (after all
the rolls) do you spend the dice. This is covered on pages 7 and 8.
Notice that nothing requires you to spend those dice on overcoming the encounter. A common early game strategy is to ignore the encounter, take a 0 or 1 Terrible Thing token (which ever is the lowest available) and load up on resource tokens for later rounds against harder encounters.
Question 3: When can a player use a feat card. Well, the answer is only in your own turn but it's not in the rulebook.
Page 7, halfway down (emphasis mine): "Before or after each re-roll, you
may perform the following actions..." One of the available actions is Perform A Feat.