If I cast something like Shaharazad or Enter The Dungeon and then cast something like Burning Wish or Research//Development from inside the sub game, is the original Shaharazad/Enter The Dungeon a legal card to bring to my hand? Since it's technically "outside" the game of magic I cast it in even though it currently exists in a zone in an active game?
Yes.
715.4. All objects in the main game and all cards outside the main game are considered outside the subgame (except those specifically brought into the subgame). All players not currently in the subgame are considered outside the subgame.715.4a. Some effects can bring cards into a game from outside of it. If a card is brought into a subgame from a main game, abilities in the main game that trigger on objects leaving a main-game zone will trigger, but they won’t be put onto the stack until the main game resumes.715.4b. A player’s main-game counters aren’t considered part of the subgame, although the player will still have them when the main game resumes. Similarly, any counters a player gets during a subgame will cease to exist when the subgame ends.
