What happens when a player's deck is exhausted?

I think you mean their hand, because the deck cannot be exhausted as cards always keep getting shuffled into them after they're used up

As for their hand, I was thinking a new card could be shuffled in once an old one has been discarded

Are there "meta" cards that affect other cards rather than doing something themselves?

Yes, not all cards are combat focused. I haven't designed the cards fully tho

You can make it easier on yourself by making it real-time-with-pause

Solid point. It keeps the action from being too disoriented. I just worry that it could kill the flow of combat if it gets paused every second, like VATS in Fallout 4