Learn English – Sport activity or a sports activity

an activity

is an on-going process over a period of time, where as

an action

is a single event or effort.

Many actions when grouped together form an activity.

Sport is a singular noun ("a sport"), whereas sports is not only the plural of sport, but also by convention refers to the general category of all sports. Example: "Tennis is a sport. Tom likes sports, but Harry only likes one sport."

Regarding phrases like "sport(s?) shoes": the way it is usually spoken, you can't hear the difference anyway, so the question is a bit esoteric. BUT the "correct" version (in AmE) would be sports shoes because we are talking about the category of sports. If the shoes are intended for one sport only, then you would say the name of the sport: "baseball shoes".