To those ends, we picked up some props last weekend that might start the basis of the apothecary kit.
It's a Boffer LARP, so there is definitely going to be combat.
It's got politics going on, so there will need to be social interplay at some level (regulated by rules or otherwise).
There will need to be scope for medics (because otherwise everyone will die too quickly and the game will be over).
Research can be long and tedious, especially in a game that operates "real time", so this sort of thing might be best handled between sessions. The same for forging weapons/armour, or other crafts.
Traditional rogue/thief tasks like picking locks, disarming traps, etc. probably require some kind of random chance element (which could also be applied to medical actions, and crafting), they could be handled through player skill, but this is a game of fantasy and escapism...the chance for players to be more than themselves for a little while. Random chance activities need the presence of a formal adjudicator (in this case a GM), and something that determines the result.
I like the idea of puzzles and riddles, but these can be hard to incorporate into a game effectively. Perhaps the randomising mechanism allows extra clues (if successful)...perhaps the mysteries are solved real time in the form of a jigsaw puzzle, or one of those gimmick puzzles found at a novelty store.
This sort of game is heavily about immersion, feeling a part of the narrative universe. The idea of randomisers (in the form of cards or dice) breaks the immersion a bit, but may be a necessity when players are forced to deal with situations they can't perform (but might be possible for their character).