For number 1, your option is correct (B). "Doggedly" is synonymous with "persistent," so "with great determination" matches that.
For number 2, your option is correct (D). "Stoutly" means "strong" and "sturdy," so "solid" would also work.
For number 3, your option is incorrect. The correct answer is B. "Boisterous" and "rowdy" both mean energetic and noisy. Since the children were bouncing off the walls, they were indeed energetic, boisterous, and rowdy.
Nick means that people can fit into only one of these four categories, which show that no one is truly happy in their lives.
Gatsby and Tom are the pursuing, Daisy is the pursued, Jordan is the busy (too busy to care about anyone else or being a good person), and Nick is tired. He's tired of having to deal with everyone else's drama and problems.
The king and duke were merely con-men pretending to be royalty. They met in Huck's raft because they were running away from the authorities who discovered their con.
They were initially strangers but upon knowing that each was a conman. Both hatched a plan to con numerous people by staging a play like Romeo and Juliet.