What does 放 mean in contexts such as 你给我放聪明点 or 你给我放老实点?
放老实点 means put in a honest way. Similarly, 放聪明点 means put in a wise way. 

放肆 means your behavior cross the line people just can not tolerate. it's kind of talk-down. 

放 is such a common word that it can be used in lots of circumstances. There will be a lot of meaning you will get from a dictionary. 

For example, 放鞭炮，放牛，放纵，放荡，绽放， 放 in these phrases means differently.