The story "How the Dog Found Himself a Master!" is a fable about a dog who goes on a journey to find a master to serve. Along the way, he encounters various animals whom he agrees to accompany and serve. However, he eventually rejects them all for different reasons.

List of Animals the Dog Agreed to Accompany and Serve:
1. The Horse: The dog agreed to serve the horse, but he quickly realized that the horse was too busy carrying the king and his retinue. The dog did not want to be ignored or forgotten, so he decided to leave.

2. The Wolf: The dog joined a pack of wolves and agreed to serve them. However, he soon discovered that the wolves were selfish and only cared about their own well-being. The dog wanted to serve a master who would care for him in return, so he left the wolves.

3. The Bear: The dog agreed to serve the bear, but he soon realized that the bear was too clumsy and always got into trouble. The dog did not want to be associated with someone who would bring danger upon him, so he decided to part ways with the bear.

4. The Lion: The dog offered to serve the lion, but he quickly realized that the lion was too proud and arrogant. The dog did not want to be subservient to someone who would not appreciate his loyalty, so he decided to reject the lion.

Reasons for Rejection:
1. Lack of Attention: The dog rejected the horse because he realized that the horse was too preoccupied with carrying the king and his retinue. The dog wanted to be noticed and valued by his master.

2. Selfishness: The dog rejected the wolves because he discovered that they only cared about themselves and did not show any concern for his well-being. The dog desired a master who would reciprocate his loyalty and care for him.

3. Clumsiness: The dog rejected the bear because he observed that the bear was constantly getting into trouble due to its clumsiness. The dog did not want to be associated with someone who would bring danger upon him.

4. Arrogance: The dog rejected the lion because he realized that the lion was too proud and arrogant. The dog desired a master who would appreciate his loyalty rather than belittle him.

In conclusion, the dog in "How the Dog Found Himself a Master!" rejected various animals as potential masters due to reasons such as lack of attention, selfishness, clumsiness, and arrogance. He was searching for a master who would appreciate and reciprocate his loyalty.