In addition to being Adroc's father and Diane's and Renton's paternal grandfather, he is also the the paternal great-grandfather of Amber Thurston and Ao Fukai, and the adoptive paternal great-grandfather of Maurice, Maeter and Linck.
Axel was a miner and mechanic during the age of trapar mining. His work lead to the discovery of the archetype of the Nirvash typeZERO LFO and a Humanoid Coralian, Eureka . His mechanical skill is legendary amongst the older generation and the military who soon adopted the LFO's for combat puproses.
His trust with the military fell heavily when the experiment with his son Adroc Thurston and Eureka failed, triggering "The First Summer of Love." Adroc vanished and was decidedly pronounced deceased. Custody of Adroc's two children, Diane (then 16) and Renton (then 4), was left to Axel. Two years later, Diane Thurston, while researching her father's work in the military, left home and vanished, and was also believed deceased.
He was then left to only take care of Renton. When Renton's high school offers him the chance to proceed into the military, Axel is outraged stating that he's lost too many family members to the military to let another go. It was that day that the Nirvash typeZERO crash lands on his home, destroying it. Eureka had come seeking the Amita drive. Although he attempts to give it to her without Renton's involvement, it is Renton who has to deliver the drive. Ironically setting Renton down the path of his family.
He helps out Gekkostate on a few occasions. When the Nirvash was upgraded to Spec 2 and needed a new ref-board, Axel was the only qualified mechanic to complete it. He finishes the board in the required time-frame and ships it to them using the lay-line due to it's fast flow and due to the military catching on to what he was doing. He had to sneak the board out of his shop and fly it like a kite, in hopes that Renton and Eureka would show up and save it from crashing to the ground.
Axel shows that he is a grumpy old man who wants Renton to be a mechanic just like him. Despite that, deep down he loves his family and wishes that nothing bad happens to them but he knows when to let go.
Rose was Axel's wife. Nothing was known about her, she was only mentioned at the end of episode 50 in the Thurston's family record book.
Following the disappearances of Adroc and Diane, Axel became protective of Renton, determined to prevent him from getting involved with the military and running off to be a hero. He always berated the boy for his dreams of joining the Gekkostate and was set on making him become a mechanic. However, he eventually decides to let Renton go with Eureka and the Gekko to let him decide his own future, and later expresses relief and pride when he says Renton was the first one in the family to return home.
Despite Eureka's past partnership with Adroc, it seems that Axel has no prior knowledge of her, but was aware that she was part of the Gekkostate and one of Holland's friends. He knew that her intention was to retrieve the Amita Drive when she first arrived. He also seemed aware of Renton and Eureka's feelings for each other; in a letter in episode 32, he asks Renton to bring Eureka with him when he decides to come home. This seems to indicate that he has always welcomed her into the family.
Renton and Eureka left Maurice, Maeter and Linck with Axel at the end of episode 50, until they came back from their journey. He is the one who reads the family record book that shows Renton's adoption of the children. He seemed to have acquired a liking to the children, even saying he was lucky to have great-grandchildren in his age.
As for his biological great-grandchildren, he never meets his great-grandson Ao because he was born and raised in an alternative universe, but he may have met his great-granddaughter Amber until the newborn girl's death.
It is obvious that Axel truly dislikes Holland for unspecific reasons, often calling him names and mocking his actions. He even went as far as to keep Diane from seeing Holland when they first started dating and blaming Holland for taking Renton away from him by inviting him to join the Gekkostate. Despite this, he is willing to help Holland, such as repairing the Nirvash's ref board after it was destroyed.