1. When exploring a tomb, do you do an explore endeavor on the space *then* pick an entry in the Captain's Booke or just declare that you are exploring the tomb and go straight for the entry?
You just declare the action, then go straight to the chosen entry. You'll always have to do an Endeavour, although it won't always be an Explore Endeavour. (Note that you do get a glory from succeeding at the specified Endeavour, unlike the occasional Endeavour you get from other things in the Captain's Booke - spoiler for where these are).
Spoiler (click to reveal)
Some Uncharted Waters cards send you to Booke entries which specify an Endeavour roll).
2. Can you transfer relics from your ship to a warehouse? I would assume that they are useless there, but it would free up a hold space
No, relics must be kept on a ship. You can throw them overboard if you want the hold space for something else in the same way as you can ditch additional goods. (In the case of relics, do the same as if the holding ship sinks - mark it as LOST and put it in the storage chest). But you can't offload them to a warehouse or your treasure room.
3. The artifacts say they don't refresh during winter. I'm guessing they are one time use per game not one time use per campaign, but wanted to be sure.
Yeah, tablets start each game refreshed, and become exhausted when used. The only way they return to being active is at the start of the next game, or if they're raided out of your treasure room by another player.