I recently had a company do a background check -- wanted me to work for them instead of private contractor. Am on good terms with them. Friends with manager's son, who also works there (lg. apt. complex-- renovating all apts.) If I ask them, nicely, can I get info. They received? Is it legal for them to do so?
Is this a bad idea all the way around?

You can ask. There is nothing illegal about them giving you info they obtained on you.