Do you keep this tile for the rest of the game ie you can build twice every turn? Or do you discard it?
You must turn in the Crane in order to perform a second Investment, so no you cannot build twice every turn - that would be way too powerful!
Alternatively can you turn in the tile to receive a Emperor Medal.
Also do joker tiles stay with you throughout the game?
Yes. These are kept - unless of course your discard them after having made a performance (which would be legal, but foolish!)