Cold One Chariot has "X" for a shield where "X" equals the number of developments. If someone were to play this card in a zone with no developments then the shield would be zero. Does that card get destroyed at that time? Same with cards that deal a -1 hit point to a unit. If this -1 is applied to a unit with a shield of 1 then that card is destroyed at that time correct?

Thanks.

Yes and yes.

Cfr. rulebook:

Q: If a unit has 0 hit points and no damage tokens, is it destroyed?
A: Yes, it is destroyed immediately because it has damage equal to its hit points.