If you're looking for an advanced strategy, the game comes down to basically one question: "Will I get card X before the game ends?"
These answer to this is based on two things, first, is the odds of you getting any particular card you want, and second, the likelihood of getting it 'in time'.
Getting a card you want is fairly simple, it's the odds it's still in the draw pile times 50%. When the game starts you have the best chance of it being in the draw pile, as opposed to already in your opponent's hand. You need to read your opponent to get at the first part of this, but for any strategy for early game, it's probably safe to assume slightly less than 50% chance of getting any card.
For the second bit, namely playing things 'in time' the rules allow counting the remaining cards. This needs to be combined with the likelihood that either you or your opponent will draw from the discard piles instead of the draw pile, thus slowing the game down. You control your draw choices, so, in effect can slow the game ending by up to twice as many turns as the remaining number of cards.
As for rules of thumb, which there aren't many because the game is quite tactical, I'd say that calculating the expected value of remaining probabilities for cards undrawn, and assuming that your opponent's hand is 75% full of the cards you need is the safe way to go.
Also, don't reveal your intentions (start expeditions) until you absolutely have to, as this increases the chance that your opponent will start/discard cards you need before you do. In this way it's not about what cards to have before starting, it's about your opponent's reveal.
So, specific answers:
- Draw until odds are less than 50%. If you have 3 cards, and 10 remain in the deck, assume your opponent has four of them, so there are 3 remaining. With proper math probabilities, you need to draw 3 times to have a better than 50% chance of getting one of the those 3 cards.
- This depends on when in the game you are. 15 points is probably too high early on, but I wouldn't start an expedition with 15 until I know if my opponent is or isn't competing in it (and they don't have hand room to block me on it)
- Probably not. Even early game.
- Depends on when in the game it is. Early on you can try for cornering the market early on it, before it's revealed. However, it's probably the first expedition you start, getting those cards out early so you can keep the next one secret (where there will be more competition).