The Remnant Cards are supposed to be placed on designated spaces on the board facedown at the start of the game. When you encounter those spaces, do you encounter the Remnant Cards, and then encounter the space as normal? What happens when there are no more Remnant Cards on that Space? Do you add more Remnant Cards to that space?

I don't have the rules here but read carefully them again and I think you'll get it.

As I remember it:

They are turned when you land on the space (so you'll know what card it is before choosing to encounter a character who might be present).

They are treated just like regular adventure cards so if there's one Remnant card on a Draw one card-space you do not draw any additional cards when encountering the space.

Once they have been discarded/picked up etc no more Remnant cards are placed. (they represent remnants from the old world...)

Just to make an obvious point. Remember that the cards are encountered face up and therefore treated as thus. If you play as the prophetess you can't use your ability to exchange the card but if you play the philosopher you can (and my best guess here is that you draw an adventure card as you discard the cards together).