As per the Patron Saint Index, there is no patron saint of horse racing. However, there are patrons of equestrians:
Anne
George
James the Greater
Martin of Tours
There is no particular saint for equestrian events but there are other patron saints for horse men, riders, equestrians, horse women:AnneGeorgeJames the GreaterMartin of Tours
There is no patron saint of auto racing or formula one.
There is no patron saint of motorsport. There are several patron saints of motorists, however:BalthasarBlaiseCasparChristopherElijah the ProphetFrances of RomeMelchiorSebastian of Aparicio
I don't know if runners have their own saint, but St. Sebastian is the Patron Saint of Athletes.
There is no patron saint of dog sledding.
Patron saints of horse men, riders, equestrians, horse women:AnneGeorgeJames the GreaterMartin of Tours
There is no particular saint for equestrian events but there are other patron saints for horse men, riders, equestrians, horse women:AnneGeorgeJames the GreaterMartin of Tours
There is no patron saint of dressage. However, the following are considered as patron saints of horse men, riders, equestrians, horse women: •Anne •George •James the Greater •Martin of Tours
The Patron Saint Inex lists no patron saints of bears.
A patron saint is supposed to be a protector in the Catholic religion. Patron Saint Angelico is the patron saint of artists. A patron saint is a protector of life.
As per the Patron Saint Index, there is no patron saint of wrestlers. However, Saint Sebastian is the patron saint of all athletics and athletes.
Saint Lillian is not a patron saint as per the Patron Saint Index.