I do know that 'any' is used for questions and negations, nevertheless sentences like 'Parma loaned Laursen out before he would play any games for the club.' make me wonder if this sentence would work to say that he would practice every sport available at the time in his school or whatever place he was.
Probably the most used sentences would be 'He used to play every sport or he used to play every single sport or sports' but I would like to know if my proposed sentence is right.
He used to play any sport.
Your proposed sentence makes sense, and the use of used to implies that it is no longer the case. Most likely, the sentence would continue with an explanation of why he no longer plays any sport:
He used to play any sport, but now he’s only interested in football.
He used to play any sport, but then he left school and got a job.
The implication is that his former habit was to play any sport that he could find, or that he would join in any game that started around him.
With a different verb this may even be a common construction:
He used to read any book[, but now he doesn’t have the time]
He used to read any book[, but now he’s only interested in science fiction]
I actually think that the use of every for sport may be less natural, because it implies that he plays every single sport, and in most situations there are more sports available to play than there is time available to play them.
Compare
I will eat anything
meaning “I am not picky about what I eat”, vs
I will eat everything
meaning either “I will eat some of each thing that is available” or “I will eat all of the food available”.