1) When you play a card from your hand, do you not draw another? Is your hand only refilled at the end of a day? You have a hand of 5 to work with, and I believe 4 phases per day.
You do not redraw cards during the day, only at the beginning of a new day. You will have five cards and four instances to use them during each day, leaving you with one at the end of each day (barring the effects of prior Destinations that increased your hand size).
2) If you have a card you cannot use, such as the one that can only be used on day 1, can you still play it to essentially do nothing (since it is worthless and reduces one of your choices, so hopefully you don't have 2 of them)?
There is only one card that can only be used in the first roundabout (there isn't a card that is restricted to the first day) and it can only be played as the first card for the day. All other cards can be played normally, and their effects must be performed.
3) For the cards that have language such as reroll 3 runners, change 1 runner to any face, or reroll 1 runner in each roundabout, is this any players' runners, or just yours?
Unless the verbiage on a card specifically states "your runner(s)", then you can choose any player's runner(s) - including your own.
4) When you "place and/or remove 1 direction token", is this referring to a single roundabout, since you may have to remove someone else's token if one is present where you want to place, or can you actually add 1 of your tokens to a roundabout (potentially removing someone else's) and also remove 1 of your tokens from a different roundabout?
We like flexibility, so this can be across multiple roundabouts if you choose.
Thanks for playing!!