Yes, we're hard at work on getting the rules in line across the board, but it's taking a bit longer than expected. Regardless, the website is now and will remain the most up-to-date core rules, at least.
1. Correct. When you spend an adjective to succeed, you cross it off or erase it: it's gone. When you go fishing in a scene, you pick up whatever adjective(s) the Narrator sets as available. (If the Narrator likes to improv, it could be a conversation between the player and Narrator about what adjectives are available. Either way, only the Narrator sets the refresh number, however, usually from zero to three.)
2. Indeed, the starting adjectives may be seen as "core" to who your character is, but in essence, this game says they are not: it assumes characters will fundamentally change by the session's end. Maybe not who they are, but certainly how their relationship with the world and other characters drives them.
In the core rules, only the descriptor is core to the character, though it's perfectly reasonable to create a Variation (our term for hacks, mods, or house rules) where one or two adjectives are part of that descriptor. That'd be the easiest way to ensure some adjectives don't go away, but then you have to consider if someone can double up on adjectives that are in their descriptor. Can a Gruff Detective pick up "Gruff" from Fishing, or as a setback? Seems like it'd be okay: you just spend the Gruff adjective and then it's gone, then you could spend any adjective to trigger Gruff Detective again (as long as you have at least one adjective to spend, that is!). Nothing game breaking in there, but just something to consider.
In the example of Conan, sure maybe he isn't Mighty any more, but he's now a Radiant Hyborean Warrior, which to me suggests we're taking a clear step towards Conan the King. He could always pick up a Mighty axe at some point, too. Either way, both those scenarios suggest something changed him or his situation, and that's precisely what DeScriptors is trying to highlight!
Thank you for such great questions! Don't hesitate to keep firing away as they crop up!