Here are the solutions for all of the Royal Train Station Challenges. You must first beat both the ship and blimp levels so you unlock more dolls for use.
Challenge 1 Empty The Lounge
Fan-tastic: Use Clerk-Of-The-Works Jasper (The doll with the wrench) and use his ability on the fan outside the lounge. Then enter and talk to the server.
Unauthorized entry: First gain control of Key Master Dawson (the doll with the key on his head) and then control The Widow Chastity (The blonde female doll outside the lounge) Seduce the guard at the entrance and then use the Key doll to open the entrance. Then talk to the server.
The Fug of War: Do the same thing as unauthorized entry but also control Meriwether Malodor (the doll that has gas problems) Enter the lounge as Meriwether and use his ability in the room.
Challenge 2 Sort the Lines
Line Assembly Production: Simply use Charlie's stacking ability to move the dolls in straight lines. Put the smallest dolls in the very leftmost lane. Put the medium dolls to the right of the smallest dolls, and finally the largest dolls to the right of the medium dolls. When the line is formed correctly you will see the ticket sign light up. Also make sure that when you line up the dolls you start the line in the very first square in each lane.
Gust of Order: Once again gain control of Meriwether Malodor and use his ability to disperse the ticket line. Then gain control of Strike Leader Chesley. Use his ability and the dolls will automatically get in line.
Uppercut Shuffle: This is by far the easiest to accomplish. Simply control the Uppercut doll and use his ability on everyone in the line. They will automatically start lining up properly.
Challenge 3 Wind the Clock
All the dolls that are needed for this challenge are all at the beginning of the train station area.
A clockwork pelican: Gain control of a pelican doll and fly up to the gears showed in the cut-scene. Then use Charlie to wind up the clock.
A Gentleman's Solution: In the circular area of the train station there should be pictures of blimps on the walls. On one of the walls is a small entrance Charlie to enter a hidden room. There should be a lever for you to open up a door. As soon as you exit the door there should be a group of 3 dolls to your right. Gain control of the doll that has the hand shake ability. Go back into the hidden room and shake the white hand that is sticking out of the wall.
Wind Under Your Wings: For this challenge you will need to gain control of the Balloon Doll and Engineer Cross (The conductor with the striped hat) Engineer Cross is next to the blimp train and the Balloon doll should be in the circular portion of the train station entrance. Once you have control of both dolls head towards the hole in the wall used in A Gentleman's Solution. You should notice two golden fans above the hole. Place Engineer Cross in the middle of the two fans and use his ability. While Engineer Cross is blowing hot air use the balloon doll's ability to fly up to the winding gear. Finally, use Charlie's stacking ability to again wind the clock gear.
After all of this you should have the Solution Sleuth Achievement.