Most other board games allow you to either slow down or (as in Talisman) pause the game when the computer's turn is running in order to read a drawn card or the like.

The computer in WAG runs REALLY fast, and even though the log shows what card an AI player has drawn, there is no way to stop the game and read the text on the card itself.

Does anyone know of a way to slow down the AI or pause the game?

Not yet (or ever xD), but you can check the actions done in the right corner. Once you are familiar with the game is enough to get what has been going on.

Is there a way to see the cards in an opponents hand on your own turn? My main issue was when Tris drew a card after completing a quest and there was a TON of non-color text on it and I have no way to see the effects it has.

All the cards that the other player use or get are shown in the right bar. If you hover the cursor over it the card stands out.
Right now I think it´s the only way in the multiplayer mode.